Algebra Terms and Definitions

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key distinctionexamplesrelated concepts

An equation makes a claim that can be tested — true, false, or conditionally true. An expression like

3x + 2

makes no claim and cannot be solved. The presence of

=

is what turns a mathematical phrase into a solvable condition.

What an Equation Is

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Expression

A mathematical phrase built from numbers, variables, and operations that represents a quantity but makes no assertion about equality.

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key distinctionexamplesrelated concepts

Expressions are evaluated or simplified. Equations are solved. The expression

2x + 1

can be factored, expanded, or evaluated at specific values of

x

, but asking "what is

x

?" makes no sense without an equation. Placing

2x + 1 = 9

creates the condition that gives

x

a definite value.

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Variable

A symbol, typically a letter, that represents an unknown quantity or a quantity that can change.

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usageexamplesrelated concepts

In equations, variables stand for unknowns: the goal is to find which values make the equation true. In expressions and functions, variables represent inputs that can take a range of values. Convention reserves

x

y

z

for unknowns,

a

b

c

for constants, and

n

k

i

for integers.

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Solution

A value of the variable that makes both sides of an equation equal when substituted.

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verificationterminologyrelated concepts

A candidate is confirmed as a solution by direct substitution. For

2x + 1 = 9

, substituting

x = 4

gives

2(4) + 1 = 9

, which is true. Substituting

x = 3

gives

2(3) + 1 = 7 \neq 9

, so

x = 3

is not a solution.

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Solution Set

The collection of all values that satisfy an equation, written in set notation.

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notationexamplesrelated concepts

Curly braces list elements:

\{-2, 2\}

for

x^2 = 4

. Set-builder notation describes conditions:

\{x \in \mathbb{R} : 2x + 1 = 9\} = \{4\}

. The empty set

\emptyset

indicates no solutions exist.

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Extraneous Solution

A value that satisfies a transformed equation but not the original, introduced by non-reversible algebraic steps.

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causespreventionrelated concepts

Squaring both sides:

x = 3

becomes

x^2 = 9

, admitting the false solution

x = -3

. Clearing denominators: multiplying by an expression that equals zero at certain values. These operations expand the solution set because they cannot be undone uniquely.

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Conditional Equation

An equation that is true for specific values of the variable and false for all others.

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propertiesexamplesrelated concepts

Most equations encountered in algebra are conditional. Solving means finding the finite set of values where the equation holds. The number of solutions depends on the equation's degree and type.

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Identity

An equation that holds true for every permissible value of the variable.

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key distinctionexamplesrelated concepts

Identities are not solved — they are verified. They express structural equivalences between expressions rather than constraints on unknowns. Expanding, factoring, or applying known rules confirms them.

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Contradiction

An equation that is false for every value of the variable — its solution set is empty.

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recognitionexamplesrelated concepts

A contradiction reveals itself during simplification: the variable terms cancel, leaving a false numerical statement. This means no value of the variable can rescue the equation.

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Equivalent Equations

Two or more equations that share exactly the same solution set.

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safe operationsrisky operationsrelated concepts

Adding or subtracting the same quantity on both sides. Multiplying or dividing both sides by a nonzero constant. These reversible operations always preserve equivalence.

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Algebraic Equation

An equation built entirely from variables, constants, and the operations of addition, subtraction, multiplication, division, and integer exponentiation.

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scopeclassification by degreerelated concepts

Algebraic equations exclude transcendental functions: no

\sin

\cos

\ln

e^x

. Equations involving those functions are classified separately as trigonometric, logarithmic, or exponential equations.

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Degree of an Equation

The highest power of the variable that appears after the equation is cleared of fractions and fully simplified.

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significanceexamplesrelated concepts

Degree determines the maximum number of solutions and dictates which solving techniques are available. Linear (degree 1) always has one solution. Quadratic (degree 2) has at most two. A polynomial of degree

n

has at most

n

roots in

\mathbb{C}

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Standard Form

The conventional way to write an equation with all terms collected on one side, arranged by descending powers of the variable, equal to zero.

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by equation typepurposerelated concepts

Linear:

ax + b = 0

Quadratic:

ax^2 + bx + c = 0

General polynomial:

a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0

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Coefficient

The numerical factor that multiplies a variable or a power of a variable in a term of an expression or equation.

Quadratic Equations Quadratic Equations Page

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typesexamplesrelated concepts

Leading coefficient: the coefficient of the highest-degree term. Free coefficient (constant term): the term with no variable attached. In

3x^2 - 7x + 2

, the leading coefficient is

3

, the coefficient of

x

-7

, and the free coefficient is

2

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Discriminant

For the quadratic equation

ax^2 + bx + c = 0

, the discriminant is

\Delta = b^2 - 4ac

Quadratic Equations

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what it determinesexamplesrelated concepts

\Delta > 0

: two distinct real solutions

\Delta = 0

: one repeated real solution

\Delta < 0

: no real solutions (two complex conjugate solutions)

The discriminant settles the question of how many real roots exist without computing them. It appears under the radical in the quadratic formula:

x = \frac{-b \pm \sqrt{\Delta}}{2a}

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Domain Restriction

A value of the variable that must be excluded from consideration because it makes an expression undefined.

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common causesrole in equationsrelated concepts

Division by zero:

\frac{1}{x-3}

excludes

x = 3

. Even roots of negatives:

\sqrt{x}

requires

x \geq 0

. Logarithms of non-positives:

\ln(x)

requires

x > 0

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Absolute Value

The distance of a number from zero on the number line:

|x| = x

x \geq 0

, and

|x| = -x

x < 0

Absolute Value Equations

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propertiesin equationsrelated concepts

Always non-negative:

|x| \geq 0

for all

x

|x| = 0

only when

x = 0

|ab| = |a| \cdot |b|

|a + b| \leq |a| + |b|

(triangle inequality)

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Roots

(20 items)

Root

The

n

th root of

b

is a value

a

such that

a^n = b

, written

\sqrt[n]{b} = a

What is a Root Connection to Powers

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intuitionexamplesrelated concepts

A root reverses exponentiation. If raising

a

to the

n

th power produces

b

, then the

n

th root of

b

recovers

a

. The radical symbol

\sqrt[n]{\phantom{x}}

denotes this operation: the small number

n

is the index, and the expression underneath is the radicand.

Roots undo powers

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Radical

The symbol

\sqrt[n]{\phantom{x}}

used to denote a root operation, consisting of the radical sign, an index, and a radicand.

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notationexamplesrelated concepts

The radical symbol has three parts:

\sqrt[n]{b}

n

— the index (which root to take). Omitted when

n = 2

b

— the radicand (the expression under the radical).

The horizontal bar extending over the radicand is called the vinculum. It acts as a grouping symbol:

\sqrt{2 + 7}

means the square root of

9

, not

\sqrt{2} + 7

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Index

The positive integer

n

\sqrt[n]{a}

that specifies which root is being taken.

Even Index Radicals Odd Index Radicals Sign Behavior Summary

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key distinctionexamplesrelated concepts

Even indices (

n = 2, 4, 6, \ldots

) require non-negative radicands and return non-negative values. Odd indices (

n = 3, 5, 7, \ldots

) accept all real radicands and preserve sign.

This single distinction controls domain restrictions, absolute value behavior during simplification, and whether extraneous solutions can arise.

Even indices restrict inputs; odd indices accept all real numbers

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Radicand

The expression placed under the radical sign in

\sqrt[n]{a}

; the value

a

from which the root is extracted.

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propertiesexamplesrelated concepts

The radicand determines whether a radical has a real value:

$\geq 0$ for a real result

When simplifying, the radicand is factored to extract perfect powers. A radicand containing a variable creates a radical function with domain restrictions governed by the index.

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Square Root

The second root of

a

: the non-negative value

b

such that

b^2 = a

, written

\sqrt{a}

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propertiesexamplesrelated concepts

$a \geq 0$ in the reals
$\sqrt{a^2} = |a|$ for all real $a$
$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ when $a, b \geq 0$
$(\sqrt{a})^2 = a$ for $a \geq 0$

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Cube Root

The third root of

a

: the value

b

such that

b^3 = a

, written

\sqrt[3]{a}

Principal Root Convention Identity for Even Roots

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propertiesexamplesrelated concepts

$\sqrt[3]{-a} = -\sqrt[3]{a}$
$\sqrt[3]{a^3} = a$ for all real $a$ (no absolute value)

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Principal Root

The unique non-negative root returned by the radical symbol

\sqrt[n]{a}

when

n

is even.

Properties

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key distinctioncommon errorsexamplesrelated concepts

Both

5

and

-5

square to

25

, but

\sqrt{25} = 5

only. The radical symbol returns one value — the non-negative root — so that it behaves as a function. When both roots are needed,

\pm\sqrt{a}

is written explicitly.

For odd indices, no convention is needed: every real number has exactly one real cube root, fifth root, etc.

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Rational Exponent

An exponent of the form

\frac{m}{n}

where

a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Rational Exponents

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intuitionexamplesrelated concepts

Every radical can be written as a fractional power. The denominator becomes the index; the numerator becomes the power applied to the base. This is not a separate operation — it is exactly the same operation in a different notation.

Exponent form often simplifies algebraic manipulation because the laws of exponents (add, subtract, multiply exponents) work with fractions just as they do with integers.

Unit Fraction Exponents General Rational Exponents Laws of Exponents with Rationals

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Product Rule (Radicals)

\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}

, provided the index matches and radicands satisfy domain restrictions.

Radical Rules

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propertiesexamplesrelated concepts

Works in both directions:

$\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$
$\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6$

Restrictions for even index: both

a \geq 0

and

b \geq 0

required in the reals. Attempting

\sqrt{-2} \cdot \sqrt{-8} = \sqrt{16}

is invalid.

Derives from the exponent law

(ab)^{1/n} = a^{1/n} \cdot b^{1/n}

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Quotient Rule (Radicals)

\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

, with

b \neq 0

and domain restrictions for even index.

Radical Rules

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propertiesexamplesrelated concepts

Works in both directions:

$\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4}$
$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$

For even index:

a \geq 0

and

b > 0

required.

Derives from

(\frac{a}{b})^{1/n} = \frac{a^{1/n}}{b^{1/n}}

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Power Rule (Radicals)

\sqrt[n]{a^m} = a^{m/n}

, equivalently

(\sqrt[n]{a})^m = a^{m/n}

Radical Rules

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propertiesexamplesrelated concepts

The exponent

m

becomes the numerator, the index

n

becomes the denominator. Either order — root first or power first — gives the same result.

When

m

and

n

share a common factor, the radical simplifies by cancellation:

\sqrt[6]{a^4} = a^{4/6} = a^{2/3} = \sqrt[3]{a^2}

For even indices with variables, absolute values may be needed:

\sqrt{x^2} = |x|

, not

x

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Simplest Form

A radical is in simplest form when no perfect power remains under the radical, no fraction appears under the radical, and no radical appears in a denominator.

What Simplest Form Means Factoring Out Perfect Powers

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conditionsexamplesrelated concepts

Three conditions must hold simultaneously:

1. No perfect power factor in the radicand. Every factor that can be extracted has been extracted.

2. No fraction under the radical.

\sqrt{\frac{3}{4}}

is rewritten as

\frac{\sqrt{3}}{2}

.

3. No radical in a denominator.

\frac{1}{\sqrt{3}}

is rationalized to

\frac{\sqrt{3}}{3}

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Perfect Square

An integer that equals

n^2

for some integer

n

0, 1, 4, 9, 16, 25, 36, \ldots

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propertiesexamplesrelated concepts

$\sqrt{n^2} = n$
$x^{2k}$ is a perfect square for any positive integer $k$

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Perfect Cube

An integer that equals

n^3

for some integer

n

0, 1, 8, 27, 64, 125, \ldots

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propertiesexamplesrelated concepts

$\sqrt[3]{n^3} = n$
$(-3)^3 = -27$ , so $-27$ is a perfect cube
$x^{3k}$ is a perfect cube for any positive integer $k$

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Rationalization

The process of rewriting an expression so that no radical appears in the denominator.

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methodsexamplesrelated concepts

For monomial denominators, multiply numerator and denominator by the radical:

\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

For binomial denominators, multiply by the conjugate:

\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \sqrt{5} - 2

For higher-index radicals, multiply to complete a perfect power:

\frac{1}{\sqrt[3]{4}} = \frac{\sqrt[3]{2}}{2}

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Like Radicals

Radical expressions that share the same index and the same radicand.

Operations

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key distinctionexamplesrelated concepts

Like radicals can be added and subtracted by combining coefficients. Unlike radicals cannot be combined.

3\sqrt{5}

and

7\sqrt{5}

— like (same index

2

, same radicand

5

)

\sqrt{3}

and

\sqrt{5}

— unlike (different radicands)

\sqrt{2}

and

\sqrt[3]{2}

— unlike (different indices)

Radicals that appear unlike may become like after simplification:

\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}

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Conjugate

For a binomial

a + b\sqrt{c}

, its conjugate is

a - b\sqrt{c}

. Their product eliminates the radical:

(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c

Operations

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propertiesexamplesrelated concepts

The product of conjugates is a difference of squares, which eliminates the radical term:

(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b

This makes conjugates the essential tool for rationalizing binomial denominators. The same algebraic identity appears in factoring and in working with complex numbers.

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Radical Equation

An equation in which the variable appears under a radical sign.

Basic Strategy Why Isolation Comes First Verification Process

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strategyexamplesrelated concepts

Four-step solving process:

1. Isolate the radical on one side
2. Raise both sides to the power matching the index
3. Solve the resulting equation
4. Check every solution in the original equation

Isolation must come first. Squaring before isolating leaves the radical intact and creates a harder equation. Step 4 is mandatory — raising to a power can introduce extraneous solutions.

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Extraneous Solution

A value that satisfies a transformed equation but not the original, introduced by a non-reversible algebraic step.

Square Root Function Cube Root Function Domain of Radical Functions Transformations

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why they appearexamplesrelated concepts

Squaring both sides is not reversible. Both

5

and

-5

square to

25

, so squaring erases sign information. If the original equation requires a radical to equal a negative value, squaring hides this impossibility.

Similarly, clearing denominators in rational equations can introduce values that zero out the original denominator. Any non-reversible step — squaring, cubing with even-degree terms, multiplying by a variable expression — carries this risk.

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Radical Function

A function of the form

f(x) = \sqrt[n]{g(x)}

, where the radicand contains a variable.

Functions

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propertiesexamplesrelated concepts

Domain depends on the index:

$g(x) \geq 0$ for the domain

The parent square root function

f(x) = \sqrt{x}

has domain

[0, \infty)

and range

[0, \infty)

. Its graph starts at the origin and rises, concave down.

The parent cube root function

f(x) = \sqrt[3]{x}

has domain and range

(-\infty, \infty)

. Its graph has an S-curve through the origin with odd symmetry.

Radical functions are inverses of power functions.

\sqrt{x}

undoes

x^2

(restricted to

x \geq 0

);

\sqrt[3]{x}

undoes

x^3

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Logarithms

(15 items)

Logarithm

The exponent to which a base

a

must be raised to produce

b

\log_a(b) = c

means

a^c = b

Logarithms

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intuitionexamplesrelated concepts

A logarithm reverses exponentiation. If

a^c = b

, then

\log_a(b) = c

. The logarithm extracts the exponent. Every logarithmic statement converts to an equivalent exponential statement and vice versa — the two forms encode identical information.

Two values hold for every valid base:

\log_a(1) = 0

because

a^0 = 1

, and

\log_a(a) = 1

because

a^1 = a

. The inverse identities

\log_a(a^x) = x

and

a^{\log_a(x)} = x

express that logarithms and exponentials undo each other.

What is a Logarithm Inverse Identities

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Base (of a Logarithm)

The number

a

\log_a(b)

; must satisfy

a > 0

and

a \neq 1

Logarithms

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restrictionsexamplesrelated concepts

The base must be positive and not equal to one:

$a > 0$ : a negative base produces complex or undefined values for most exponents
$a \neq 1$ : since $1^c = 1$ for all $c$ , no exponent distinguishes one output from another
$0 < a < 1$ is valid: produces a decreasing logarithmic function
$a > 1$ : produces an increasing logarithmic function

The base determines the direction of monotonicity, which controls whether inequality direction is preserved or reversed.

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Argument (of a Logarithm)

The number

b

\log_a(b)

; must satisfy

b > 0

Logarithms

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restrictionsexamplesrelated concepts

The argument must be strictly positive. Since

a^c > 0

for any positive base

a

and any real exponent

c

, no real exponent can produce zero or a negative result.

$\log_a(0)$ is undefined — no solution to $a^c = 0$ exists
$\log_a(-5)$ is undefined — no real $c$ satisfies $a^c = -5$

For composite arguments like

\log_2(x - 3)

, the entire expression must be positive:

x - 3 > 0

, giving

x > 3

. This determines the domain of logarithmic functions and creates the possibility of extraneous solutions when solving equations.

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Common Logarithm

The logarithm with base

10

, written

\log(x)

\log_{10}(x)

Common & Natural

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notationexamplesrelated concepts

Written

\log(x)

without a subscript. Some textbooks and programming languages use

\log

to mean the natural logarithm instead — context determines which convention applies. When ambiguity is possible,

\log_{10}(x)

makes the base explicit.

Calculators typically label the base-10 key as LOG.

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Natural Logarithm

The logarithm with base

e \approx 2.71828

, written

\ln(x)

\log_e(x)

Common & Natural

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notationkey distinctionexamplesrelated concepts

Written

\ln(x)

, read "natural log of

x

." The abbreviation comes from the Latin logarithmus naturalis.

Calculators label the base-

e

key as LN. In many programming languages and advanced mathematics texts,

\log

without subscript means

\ln

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Euler's Number (e)

The irrational constant

e \approx 2.71828

, defined as

\lim_{n \to \infty} (1 + 1/n)^n

Common & Natural

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propertiesexamplesrelated concepts

$e = 2.718281828459045\ldots$
$\ln(x) = \log_e(x)$
$\frac{d}{dx} e^x = e^x$
$\lim_{n \to \infty} (1 + 1/n)^n = e$
$e = \sum_{n=0}^{\infty} \frac{1}{n!}$

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Product Rule (Logarithms)

\log_a(xy) = \log_a(x) + \log_a(y)

— the logarithm of a product equals the sum of the logarithms.

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derivationexamplesrelated concepts

Let

\log_a(x) = m

and

\log_a(y) = n

. Then

x = a^m

and

y = a^n

xy = a^m \cdot a^n = a^{m+n}

\log_a(xy) = m + n = \log_a(x) + \log_a(y)

The rule derives from the exponent law

a^m \cdot a^n = a^{m+n}

. Multiplication inside the logarithm becomes addition outside.

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Quotient Rule (Logarithms)

\log_a(x/y) = \log_a(x) - \log_a(y)

— the logarithm of a quotient equals the difference of the logarithms.

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derivationexamplesrelated concepts

Let

\log_a(x) = m

and

\log_a(y) = n

. Then

x/y = a^m / a^n = a^{m-n}

\log_a(x/y) = m - n = \log_a(x) - \log_a(y)

Special case:

\log_a(1/x) = -\log_a(x)

, since

\log_a(1) = 0

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Power Rule (Logarithms)

\log_a(x^n) = n \cdot \log_a(x)

— exponents inside the argument become coefficients outside.

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derivationexamplesrelated concepts

Let

\log_a(x) = m

, so

x = a^m

. Then

x^n = (a^m)^n = a^{mn}

\log_a(x^n) = mn = n \cdot \log_a(x)

Works for any real exponent

n

, including fractions and negatives:

\log_a(\sqrt{x}) = \log_a(x^{1/2}) = \frac{1}{2} \log_a(x)

\log_a(x^{-3}) = -3 \log_a(x)

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Change of Base Formula

\log_a(x) = \frac{\log_b(x)}{\log_b(a)}

— converts a logarithm from one base to any other.

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derivationexamplesrelated concepts

Let

\log_a(x) = y

, so

a^y = x

. Take

\log_b

of both sides:

\log_b(a^y) = \log_b(x)

y \cdot \log_b(a) = \log_b(x)

y = \frac{\log_b(x)}{\log_b(a)}

Special case:

\log_a(b) = \frac{1}{\log_b(a)}

— swapping base and argument gives the reciprocal.

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Monotonicity

A logarithmic function is strictly increasing when

a > 1

and strictly decreasing when

0 < a < 1

Properties

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key distinctionexamplesrelated concepts

For

a > 1

: if

x_1 < x_2

, then

\log_a(x_1) < \log_a(x_2)

. Inequality direction is preserved.

For

0 < a < 1

: if

x_1 < x_2

, then

\log_a(x_1) > \log_a(x_2)

. Inequality direction is reversed.

This is the single most important property when solving logarithmic inequalities. Taking logarithms of both sides preserves or reverses the inequality depending entirely on the base.

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One-to-One Property

\log_a(x) = \log_a(y)

, then

x = y

. No two distinct inputs produce the same output.

Properties

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strategyexamplesrelated concepts

This property justifies a key equation-solving technique: when both sides of an equation are logarithms with the same base, set the arguments equal and solve the resulting algebraic equation.

\log_a(M) = \log_a(N) \implies M = N

Every solution must still be checked against domain restrictions — both

M > 0

and

N > 0

must hold.

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Logarithmic Equation

An equation in which the variable appears inside the argument of a logarithm.

The Basic Shape The Vertical Asymptote Key Points Effect of Base on Shape

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strategyexamplesrelated concepts

Two main forms:

1. Logarithm equals a constant:

\log_a(f(x)) = k

— convert to exponential form:

f(x) = a^k

2. Logarithms on both sides:

\log_a(M) = \log_a(N)

— apply the one-to-one property:

M = N

When multiple logarithms appear, use the product, quotient, or power rules to condense into a single logarithm first. Every solution must satisfy domain restrictions: all original arguments must be positive.

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Logarithmic Inequality

An inequality involving a logarithmic expression, where the base determines whether inequality direction is preserved or reversed.

Inequalities

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key distinctionexamplesrelated concepts

The solving process mirrors equations with one critical addition: the base controls direction.

For

a > 1

(increasing function):

\log_a(x) > k \implies x > a^k

— direction preserved

For

0 < a < 1

(decreasing function):

\log_a(x) > k \implies x < a^k

— direction reversed

Domain restrictions apply throughout: every argument must remain positive. The final answer is the intersection of the algebraic solution with the domain.

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Logarithmic Function

The function

f(x) = \log_a(x)

with domain

(0, \infty)

, range

(-\infty, \infty)

, and vertical asymptote at

x = 0

Graphs

See details

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propertiesexamplesrelated concepts

$(0, \infty)$ — only positive inputs
$(-\infty, \infty)$ — all real outputs
$(1, 0)$ and $(a, 1)$ for every base
$x = 0$ (the $y$ -axis)
$a > 1$
$a > 1$ ; strictly decreasing when $0 < a < 1$
$g(x) = a^x$ : their graphs reflect across $y = x$

Transformations shift the asymptote and key points:

y = \log_a(x - h) + k

has asymptote at

x = h

and passes through

(1 + h, k)

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Polynomials

(27 items)

Polynomial

An expression of the form

a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

where the exponents are non-negative integers.

Polynomials

See details

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intuitionexamplesrelated concepts

A polynomial is built from three ingredients: a variable, coefficients, and non-negative integer powers. Addition, subtraction, and multiplication are allowed; division by the variable is not. The expression

3x^4 - 2x^2 + 5x - 7

qualifies. The expressions

x^{-2} + 3

\sqrt{x} + 1

, and

2^x

do not — each violates the non-negative integer exponent requirement.

Polynomials are closed under addition, subtraction, and multiplication: combining two polynomials through these operations always yields another polynomial. Division can break this —

\frac{x^2+1}{x}

is not a polynomial.

What is a Polynomial?

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Term (of a Polynomial)

A single unit within a polynomial: a coefficient multiplied by a power of the variable, such as

4x^3

-2x

7

See details

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intuitionkey distinctionrelated concepts

Every polynomial is a sum of terms. In

4x^3 - 2x + 7

, the three terms are

4x^3

-2x

, and

7

. Each term has a coefficient (numerical factor) and a degree (the exponent on the variable). The signs belong to the terms — the second term is

-2x

, not

2x

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Leading Coefficient

The coefficient of the highest-degree term in a polynomial written in standard form.

See details

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intuitionpropertiesrelated concepts

4x^3 - 2x + 7

, the leading term is

4x^3

and the leading coefficient is

4

. This single number controls the polynomial's large-scale behavior: it determines whether the graph ultimately rises or falls and how steeply. Two polynomials of the same degree with opposite leading coefficients produce mirror-image end behavior.

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Constant Term

The term with no variable factor — the value

a_0

in a polynomial, equal to

P(0)

See details

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intuitionexamplesrelated concepts

The constant term is what remains when

x = 0

. In

3x^2 - 5x + 8

, the constant term is

8

, and indeed

P(0) = 8

. Geometrically, it gives the y-intercept of the polynomial's graph. In the Rational Root Theorem, the factors of the constant term form the numerator candidates for rational roots.

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Degree (of a Polynomial)

The highest exponent appearing on the variable in any term of the polynomial.

Polynomial Degree

See details

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intuitionpropertiesrelated concepts

Degree measures a polynomial's complexity. It caps the number of roots, bounds the number of turning points to

n-1

, and determines end behavior. A constant like

5

has degree

0

. A linear expression like

2x+1

has degree

1

. The zero polynomial is a special case — its degree is left undefined or assigned

-\infty

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Monomial

A polynomial with exactly one term: a coefficient times a power of the variable, such as

5x^2

See details

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exampleskey distinctionrelated concepts

7x^3

— monomial of degree

3

-4

— monomial of degree

0

x

— monomial of degree

1

with coefficient

1

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Binomial

A polynomial with exactly two terms, such as

x + 3

x^2 - 4

See details

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exampleskey distinctionrelated concepts

x + 3

— linear binomial

x^2 - 4

— difference of squares, factors as

(x+2)(x-2)

x^3 + 8

— sum of cubes, factors as

(x+2)(x^2-2x+4)

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Trinomial

A polynomial with exactly three terms, such as

x^2 + 5x + 6

Perfect Square Trinomials

See details

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exampleskey distinctionrelated concepts

x^2 + 5x + 6

— factors as

(x+2)(x+3)

x^2 - 4x + 4

— perfect square trinomial, factors as

(x-2)^2

2x^2 + 3x - 5

— general-case trinomial requiring the ac-method

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Like Terms

Terms that share the same variable raised to the same exponent, differing only in their coefficients.

Adding Polynomials

See details

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intuitioncommon errorsrelated concepts

3x^2

and

-5x^2

are like terms — same variable, same exponent. They combine to

-2x^2

. The terms

3x^2

and

3x^3

are not like terms because the exponents differ. Combining like terms is the mechanism behind polynomial addition and subtraction.

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Root (of a Polynomial)

A value

r

such that

P(r) = 0

. Also called a zero or solution of the polynomial equation

P(x) = 0

Roots of a Polynomial

See details

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intuitionpropertiesrelated concepts

A root is an input that makes the polynomial vanish. If

P(x) = x^2 - 5x + 6

, then

P(2) = 0

and

P(3) = 0

, so

2

and

3

are roots. Each root corresponds to a factor: root

r

gives factor

(x - r)

. Geometrically, real roots are the x-intercepts of the polynomial's graph.

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Multiplicity

The number of times a root

r

appears as a factor

(x - r)^k

in the polynomial's factorization;

k

is the multiplicity.

Multiplicity

See details

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intuitionpropertiesrelated concepts

(x-2)^3(x+1)

, the root

x = 2

has multiplicity

3

and the root

x = -1

has multiplicity

1

. The total multiplicities sum to the degree. Multiplicity determines how the graph behaves at the root: odd multiplicity causes a crossing, even multiplicity causes the graph to touch the axis and turn back.

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Factoring

Writing a polynomial as a product of two or more polynomials of lower degree.

Factoring Polynomials

See details

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intuitionmethodsrelated concepts

Factoring reverses multiplication. The polynomial

x^2 + 5x + 6

factors as

(x+2)(x+3)

. The factored form reveals roots directly and simplifies equation-solving, graphing, and simplification of rational expressions. A polynomial is fully factored when no factor can be broken down further over the chosen number system.

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Greatest Common Factor (GCF)

The largest expression — numerical and variable parts combined — that divides evenly into every term of a polynomial.

GCF Factoring

See details

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intuitionexamplesrelated concepts

For

6x^3 + 9x^2 - 3x

, each term is divisible by

3x

. Factoring gives

3x(2x^2 + 3x - 1)

. The GCF takes the lowest power of each variable present across all terms and the numerical GCF of all coefficients.

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Difference of Squares

a^2 - b^2 = (a + b)(a - b)

Difference of Squares

See details

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intuitionexamplesrelated concepts

Any expression that matches the pattern "something squared minus something else squared" factors instantly into conjugate binomials. The key is recognizing the pattern: both terms must be perfect squares, joined by subtraction. A sum of squares

a^2 + b^2

does not factor over the reals.

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Perfect Square Trinomial

a^2 + 2ab + b^2 = (a + b)^2

and

a^2 - 2ab + b^2 = (a - b)^2

See details

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intuitionexamplesrelated concepts

A trinomial is a perfect square when the first and last terms are perfect squares and the middle term equals twice their product. Recognizing this pattern avoids trial-and-error factoring entirely — the factored form is immediate.

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Sum and Difference of Cubes

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

and

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Sum and Difference of Cubes

See details

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intuitionexamplesrelated concepts

Unlike squares, both the sum and the difference of two cubes factor. The pattern produces a binomial factor and a trinomial factor. The signs follow a consistent rule: the binomial matches the original sign, and in the trinomial the middle term flips sign while the last stays positive.

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Irreducible Polynomial

A polynomial that cannot be factored into polynomials of lower degree over a given number system.

Irreducible Polynomials

See details

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intuitionexamplesrelated concepts

Whether a polynomial is irreducible depends on the number system. Over the reals,

x^2 + 1

is irreducible — no pair of real-coefficient linear factors multiplies to give it. Over the complex numbers,

x^2 + 1 = (x+i)(x-i)

and is therefore reducible. The Fundamental Theorem of Algebra guarantees that every polynomial factors completely into linear factors over

\mathbb{C}

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End Behavior

The direction the graph of a polynomial heads as

x \to +\infty

and

x \to -\infty

, determined by the leading term

a_nx^n

End Behavior

See details

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propertiesexamplesrelated concepts

Controlled entirely by the degree

n

and the sign of

a_n

:

• Even degree,

a_n > 0

: both ends rise (

\uparrow\;\uparrow

)
• Even degree,

a_n < 0

: both ends fall (

\downarrow\;\downarrow

)
• Odd degree,

a_n > 0

: left falls, right rises (

\downarrow\;\uparrow

)
• Odd degree,

a_n < 0

: left rises, right falls (

\uparrow\;\downarrow

)

No other coefficient affects end behavior — for sufficiently large

|x|

, the leading term dominates.

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Turning Point

A point where a polynomial's graph changes from increasing to decreasing or vice versa; a polynomial of degree

n

has at most

n - 1

turning points.

Turning Points

See details

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intuitionexamplesrelated concepts

Turning points are local maxima and local minima — the peaks and valleys of the curve. A quadratic has exactly one (the vertex). A cubic has at most two. The bound

n-1

is a maximum, not a guarantee: the cubic

x^3

has zero turning points because it increases monotonically.

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Remainder Theorem

When

P(x)

is divided by

(x - c)

, the remainder equals

P(c)

Remainder Theorem

See details

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intuitionexamplesrelated concepts

The remainder can be found without performing division — just evaluate the polynomial at

c

. This works because

P(x) = (x - c) \cdot Q(x) + R

, and substituting

x = c

annihilates the quotient term, leaving

P(c) = R

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Factor Theorem

(x - c)

is a factor of

P(x)

if and only if

P(c) = 0

Factor Theorem

See details

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intuitionexamplesrelated concepts

A direct consequence of the Remainder Theorem: if the remainder is zero, the division is exact and

(x-c)

divides

P(x)

. This bridges roots and factors — finding a root immediately gives a factor, and confirming a factor immediately gives a root.

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Rational Root Theorem

\frac{p}{q}

(in lowest terms) is a rational root of

a_nx^n + \cdots + a_0

, then

p

divides

a_0

and

q

divides

a_n

Rational Root Theorem

See details

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intuitionexamplesrestrictionsrelated concepts

This theorem converts root-finding from open-ended guessing into a finite checklist. List all factors of the constant term (candidates for

p

) and all factors of the leading coefficient (candidates for

q

). Form every ratio

\pm p/q

. Only these values can be rational roots — test each using the Remainder Theorem.

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Descartes' Rule of Signs

The number of positive real roots of

P(x)

equals the number of sign changes in its coefficients, or less by an even number. For negative roots, apply the rule to

P(-x)

Descartes' Rule of Signs

See details

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intuitionexamplesrelated concepts

Before testing any specific value, the sequence of coefficient signs reveals how many positive and negative roots are possible. Count sign changes in the coefficient sequence of

P(x)

for positive roots, and in

P(-x)

for negative roots. Each count gives the maximum, with the actual number differing by

0, 2, 4, \ldots

(always by an even amount).

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Fundamental Theorem of Algebra

Every polynomial of degree

n \geq 1

with complex coefficients has exactly

n

roots in

\mathbb{C}

, counted with multiplicity.

Fundamental Theorem

See details

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intuitionpropertiesrelated concepts

This theorem guarantees that roots always exist — the complex number system is large enough that no polynomial of positive degree can avoid having a root. It also means every polynomial of degree

n

factors completely into

n

linear factors over

\mathbb{C}

P(x) = a_n(x - r_1)(x - r_2) \cdots (x - r_n)

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Vieta's Formulas

For

P(x) = a_n x^n + \cdots + a_0

with roots

r_1, \ldots, r_n

r_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n}

and

r_1 \cdot r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}

Vieta's Formulas

See details

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intuitionexamplesrelated concepts

Vieta's formulas relate a polynomial's roots directly to its coefficients, bypassing the need to find the roots individually. The sum of all roots equals

-a_{n-1}/a_n

. The product of all roots equals

(-1)^n a_0/a_n

. For a quadratic

ax^2 + bx + c

, this simplifies to

r_1 + r_2 = -b/a

and

r_1 \cdot r_2 = c/a

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Synthetic Division

A shorthand method for dividing a polynomial by a linear binomial

(x - c)

using only coefficients and the value

c

Synthetic Division

See details

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intuitionrestrictionsrelated concepts

Synthetic division compresses polynomial long division into a compact numerical procedure. Write only the coefficients (including zeros for missing terms), bring down the first, multiply by

c

, add down, repeat. The final row gives the quotient coefficients and the remainder — which, by the Remainder Theorem, equals

P(c)

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Polynomial Long Division

A systematic procedure for dividing one polynomial by another, producing a quotient and a remainder:

P(x) = D(x) \cdot Q(x) + R(x)

Polynomial Long Division

See details

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intuitionkey distinctionrelated concepts

The algorithm mirrors long division of integers. Divide the leading term of the dividend by the leading term of the divisor, multiply the result back through the entire divisor, subtract, and repeat with the new reduced polynomial. The process terminates when the degree of the remaining polynomial is less than the degree of the divisor.

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Exponents

(15 items)

Power

An expression

a^n

consisting of a base

a

raised to an exponent

n

Powers

See details

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intuitionexamplesrelated concepts

A power compresses repeated multiplication into compact notation. The expression

a^n

means

a

multiplied by itself

n

times when

n

is a positive integer. The word "power" refers to the entire expression — not just the exponent.

Special names:

a^2

is "

a

squared,"

a^3

is "

a

cubed." All others use "

a

to the

n

th power."

The concept extends beyond counting repetitions: zero, negative, fractional, and irrational exponents each broaden the meaning while preserving the same algebraic laws.

Definition and Terminology Notation and Conventions

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Base (of a Power)

The number

a

a^n

— the quantity being raised to a power.

Powers

See details

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key distinctionexamplesrelated concepts

Parentheses determine what counts as the base:

-2^2 = -(2^2) = -4

— base is

2

, exponent applies to

2

only

(-2)^2 = (-2)(-2) = 4

— base is

-2

, exponent applies to the entire quantity

Without parentheses, the exponent binds to the nearest symbol. The negative sign is not part of the base unless parentheses force it.

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Exponent

The number

n

a^n

that controls how the base is used — counting repetitions for natural exponents, and generalizing through zero, negative, rational, and irrational values.

Powers

See details

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progressionexamplesrelated concepts

Each exponent type extends the previous while preserving the laws:

$n = 1, 2, 3, \ldots$ ): count repeated multiplications
$n = 0$ ): $a^0 = 1$ for $a \neq 0$ , forced by the quotient rule
$n < 0$ ): $a^{-n} = 1/a^n$ , extends the pattern below zero
$n = m/k$ ): $a^{m/k} = \sqrt[k]{a^m}$ , connects to roots
$n = \pi, \sqrt{2}, \ldots$ ): defined as the limit of rational approximations

Domain restrictions tighten at each step: natural allows any base, negative excludes

0

, rational with even denominator requires

a \geq 0

, irrational requires

a > 0

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Natural Exponent

A positive integer exponent:

a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}}

for

n \geq 1

Natural Exponents

See details

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propertiesexamplesrelated concepts

$(-3)^2 = 9$
$(-3)^3 = -27$
$a^1 = a$ for any $a$ ; $1^n = 1$ for any $n$

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Zero Exponent

a^0 = 1

for any

a \neq 0

Zero Powers

See details

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why it equals oneexamplesrelated concepts

Forced by the quotient rule:

a^n / a^n = a^{n-n} = a^0

. Since

a^n / a^n = 1

, we must have

a^0 = 1

.

A pattern argument confirms:

3^3 = 27

3^2 = 9

3^1 = 3

. Each step divides by

3

. Continuing:

3^0 = 1

.

The case

0^0

is genuinely ambiguous — combinatorics assigns it

1

(empty product), analysis leaves it undefined (indeterminate form

0^0

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Negative Exponent

a^{-n} = \frac{1}{a^n}

for

a \neq 0

Negative Exponents

See details

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why reciprocalsexamplesrelated concepts

Forced by the quotient rule:

a^2 / a^5 = a^{2-5} = a^{-3}

. By cancellation,

a^2 / a^5 = 1/a^3

. So

a^{-3} = 1/a^3

.

The pattern extends the descending sequence:

3^1 = 3

3^0 = 1

3^{-1} = 1/3

3^{-2} = 1/9

.

Base cannot be zero —

0^{-n}

requires dividing by

0^n = 0

, which is undefined.

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Irrational Exponent

An exponent that cannot be expressed as a fraction, such as

\pi

\sqrt{2}

. The value

a^x

is defined as the limit of

a^r

as rational

r

approaches

x

Irrational Exponents

See details

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how it worksexamplesrelated concepts

The number

\pi

is irrational, so

2^\pi

cannot use the rational exponent definition. Instead, trap

\pi

between rational bounds and compute the corresponding powers:

2^3 = 8

2^{3.1} \approx 8.574

2^{3.14} \approx 8.815

2^{3.1415} \approx 8.824

The values converge to

2^\pi \approx 8.825

.

This requires

a > 0

— for

a \leq 0

, the sequence of rational approximations does not converge consistently.

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Product Rule (Exponents)

a^m \cdot a^n = a^{m+n}

— same base, add exponents.

See details

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derivationexamplesrelated concepts

For natural exponents:

a^m \cdot a^n

places

m

factors of

a

followed by

n

more, giving

m + n

total factors.

The rule extends to all exponent types by definition — zero, negative, rational, and irrational exponents are chosen precisely to keep this identity valid.

Bases must match:

2^3 \cdot 3^4

cannot be simplified by adding exponents.

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Quotient Rule (Exponents)

\frac{a^m}{a^n} = a^{m-n}

— same base, subtract exponents.

See details

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derivationexamplesrelated concepts

For natural exponents with

m > n

: cancel

n

common factors from numerator and denominator, leaving

m - n

factors.

When

m = n

: gives

a^0 = 1

— forces the zero exponent definition.
When

m < n

: gives negative exponents — forces the negative exponent definition.

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Power of a Power

(a^m)^n = a^{mn}

— raise a power to a power, multiply exponents.

See details

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derivationexamplesrelated concepts

(a^m)^n

means

a^m

multiplied by itself

n

times. By the product rule, this adds

m

a total of

n

times:

m + m + \cdots + m = mn

.

Note:

(a^m)^n \neq a^{m^n}

. Stacked exponents read top-down:

a^{m^n}

means

a^{(m^n)}

, not

(a^m)^n

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Power of a Product

(ab)^n = a^n \cdot b^n

— distribute the exponent across multiplication.

See details

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derivationexamplesrelated concepts

(ab)^n = (ab)(ab)\cdots(ab)

—

n

copies. Rearranging factors: all

a

's together and all

b

's together gives

a^n \cdot b^n

.

This is the exponent law behind the product rule for radicals:

\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}

(ab)^{1/n} = a^{1/n} \cdot b^{1/n}

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Power of a Quotient

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

— distribute the exponent across division,

b \neq 0