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SECTIONAlgebra12 subsectionsBROWSE ALL ↓INTERACTIVE2 TOOLSAlgebra VisualTools7 toolsJump to ↓AlgebraCalculators andSolvers29 toolsJump to ↓fREFERENCE189 ITEMSfAlgebra Formulasand Identities85 itemsJump to ↓AaAlgebra Terms andDefinitions104 itemsJump to ↓§CORE TOPICS8 SUBSECTIONSEquations5 TOPICSJump to ↓Algebraic IdentitiesJump to ↓Inequalities in Algebra5 TOPICSJump to ↓Logarithms6 TOPICSJump to ↓Polynomials5 TOPICSJump to ↓Powers and Exponents9 TOPICSJump to ↓Roots and Radicals7 TOPICSJump to ↓Sequences in Mathematics7 TOPICSJump to ↓

Algebra Visual Tools

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Interactive geometric proofs of fundamental algebraic identities. Watch (a+b)², (a-b)², (a+b+c)², and a²-b² come alive through animated dissection — see exactly why each identity holds, step by step.
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Algebra Formulas and Identities

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Formulas are the working tools of algebra -- each one encodes a pattern that appears across hundreds of problems. This reference collects every major formula with its conditions, derivations, and connections to related results, so you can find the right identity quickly and understand why it works. Use it while solving homework, preparing for exams, or building fluency with algebraic manipulation. Formulas are organized into six searchable categories: Equations (quadratic formula, discriminant, completing the square, square root property, absolute value equations and inequalities), Logarithm Rules (product rule, quotient rule, power rule, change of base, log of base, log of one, inverse cancellation properties), Identities & Factoring (difference of squares, squares and cubes of sums and differences, sum and difference of cubes, trinomial factoring, general power factorizations), Exponent Rules (product rule, quotient rule, power of a power, power of a product, power of a quotient, zero and negative exponents), Radical Rules (radical-to-exponent conversion, product rule, quotient rule, power rule, nested radicals, even and odd root identities), and Polynomial Theorems (remainder theorem, factor theorem, rational root theorem, Vieta
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Algebra Terms and Definitions

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A clear vocabulary is the foundation of every algebra course -- without it, formulas look like noise and word problems become guesswork. This glossary gives each concept a precise definition, concrete examples, and the context needed to connect it to the rest of the subject. Use it as a reference while solving homework, reviewing for exams, or bridging gaps from earlier courses. Terms are organized into six searchable categories: Equations (variables, solution sets, identities, contradictions, discriminant, absolute value), Roots and Radicals (square roots, cube roots, nth roots, simplification, rationalization, surds), Logarithms (natural log, common log, binary log, identities, equations, change of base), Polynomials (degree, factoring, division, Vieta
Equations17
EquationA statement asserting that two mathematical expressions have the same value, written with the == sign between them.Read more →ExpressionA mathematical phrase built from numbers, variables, and operations that represents a quantity but makes no assertion about equality.Read more →VariableA symbol, typically a letter, that represents an unknown quantity or a quantity that can change.Read more →SolutionA value of the variable that makes both sides of an equation equal when substituted.Read more →Solution SetThe collection of all values that satisfy an equation, written in set notation.Read more →Extraneous SolutionA value that satisfies a transformed equation but not the original, introduced by non-reversible algebraic steps.Read more →Conditional EquationAn equation that is true for specific values of the variable and false for all others.Read more →IdentityAn equation that holds true for every permissible value of the variable.Read more →ContradictionAn equation that is false for every value of the variable — its solution set is empty.Read more →Equivalent EquationsTwo or more equations that share exactly the same solution set.Read more →Algebraic EquationAn equation built entirely from variables, constants, and the operations of addition, subtraction, multiplication, division, and integer…Read more →Degree of an EquationThe highest power of the variable that appears after the equation is cleared of fractions and fully simplified.Read more →Standard FormThe conventional way to write an equation with all terms collected on one side, arranged by descending powers of the variable, equal to zero.Read more →CoefficientThe numerical factor that multiplies a variable or a power of a variable in a term of an expression or equation.Read more →DiscriminantFor the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the discriminant is Δ=b2−4ac\Delta = b^2 - 4ac.Read more →Domain RestrictionA value of the variable that must be excluded from consideration because it makes an expression undefined.Read more →Absolute ValueThe distance of a number from zero on the number line: ∣x∣=x|x| = x if x≥0x \geq 0, and ∣x∣=−x|x| = -x if x<0x < 0.Read more →
RootThe nnth root of bb is a value aa such that an=ba^n = b, written bn=a\sqrt[n]{b} = a.Read more →RadicalThe symbol xn\sqrt[n]{\phantom{x}} used to denote a root operation, consisting of the radical sign, an index, and a radicand.Read more →IndexThe positive integer nn in an\sqrt[n]{a} that specifies which root is being taken.Read more →RadicandThe expression placed under the radical sign in an\sqrt[n]{a}; the value aa from which the root is extracted.Read more →Square RootThe second root of aa: the non-negative value bb such that b2=ab^2 = a, written a\sqrt{a}.Read more →Cube RootThe third root of aa: the value bb such that b3=ab^3 = a, written a3\sqrt[3]{a}.Read more →Principal RootThe unique non-negative root returned by the radical symbol an\sqrt[n]{a} when nn is even.Read more →Rational ExponentAn exponent of the form mn\frac{m}{n} where am/n=amn=(an)ma^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m.Read more →Product Rule (Radicals)abn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}, provided the index matches and radicands satisfy domain restrictions.Read more →Quotient Rule (Radicals)abn=anbn\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}, with b≠0b \neq 0 and domain restrictions for even index.Read more →Power Rule (Radicals)amn=am/n\sqrt[n]{a^m} = a^{m/n}, equivalently (an)m=am/n(\sqrt[n]{a})^m = a^{m/n}.Read more →Simplest FormA 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…Read more →Perfect SquareAn integer that equals n2n^2 for some integer nn: 0,1,4,9,16,25,36,…0, 1, 4, 9, 16, 25, 36, \ldotsRead more →Perfect CubeAn integer that equals n3n^3 for some integer nn: 0,1,8,27,64,125,…0, 1, 8, 27, 64, 125, \ldotsRead more →RationalizationThe process of rewriting an expression so that no radical appears in the denominator.Read more →Like RadicalsRadical expressions that share the same index and the same radicand.Read more →ConjugateFor a binomial a+bca + b\sqrt{c}, its conjugate is a−bca - b\sqrt{c}. Their product eliminates the radical: (a+bc)(a−bc)=a2−b2c(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c.Read more →Radical EquationAn equation in which the variable appears under a radical sign.Read more →Extraneous SolutionA value that satisfies a transformed equation but not the original, introduced by a non-reversible algebraic step.Read more →Radical FunctionA function of the form f(x)=g(x)nf(x) = \sqrt[n]{g(x)}, where the radicand contains a variable.Read more →
LogarithmThe exponent to which a base aa must be raised to produce bb: log⁡a(b)=c\log_a(b) = c means ac=ba^c = b.Read more →Base (of a Logarithm)The number aa in log⁡a(b)\log_a(b); must satisfy a>0a > 0 and a≠1a \neq 1.Read more →Argument (of a Logarithm)The number bb in log⁡a(b)\log_a(b); must satisfy b>0b > 0.Read more →Common LogarithmThe logarithm with base 1010, written log⁡(x)\log(x) or log⁡10(x)\log_{10}(x).Read more →Natural LogarithmThe logarithm with base e≈2.71828e \approx 2.71828, written ln⁡(x)\ln(x) or log⁡e(x)\log_e(x).Read more →Euler's Number (e)The irrational constant e≈2.71828e \approx 2.71828, defined as lim⁡n→∞(1+1/n)n\lim_{n \to \infty} (1 + 1/n)^n.Read more →Product Rule (Logarithms)log⁡a(xy)=log⁡a(x)+log⁡a(y)\log_a(xy) = \log_a(x) + \log_a(y) — the logarithm of a product equals the sum of the logarithms.Read more →Quotient Rule (Logarithms)log⁡a(x/y)=log⁡a(x)−log⁡a(y)\log_a(x/y) = \log_a(x) - \log_a(y) — the logarithm of a quotient equals the difference of the logarithms.Read more →Power Rule (Logarithms)log⁡a(xn)=n⋅log⁡a(x)\log_a(x^n) = n \cdot \log_a(x) — exponents inside the argument become coefficients outside.Read more →Change of Base Formulalog⁡a(x)=log⁡b(x)log⁡b(a)\log_a(x) = \frac{\log_b(x)}{\log_b(a)} — converts a logarithm from one base to any other.Read more →MonotonicityA logarithmic function is strictly increasing when a>1a > 1 and strictly decreasing when 0<a<10 < a < 1.Read more →One-to-One PropertyIf log⁡a(x)=log⁡a(y)\log_a(x) = \log_a(y), then x=yx = y. No two distinct inputs produce the same output.Read more →Logarithmic EquationAn equation in which the variable appears inside the argument of a logarithm.Read more →Logarithmic InequalityAn inequality involving a logarithmic expression, where the base determines whether inequality direction is preserved or reversed.Read more →Logarithmic FunctionThe function f(x)=log⁡a(x)f(x) = \log_a(x) with domain (0,∞)(0, \infty), range (−∞,∞)(-\infty, \infty), and vertical asymptote at x=0x = 0.Read more →
PolynomialAn expression of the form anxn+an−1xn−1+⋯+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 where the exponents are non-negative integers.Read more →Term (of a Polynomial)A single unit within a polynomial: a coefficient multiplied by a power of the variable, such as 4x34x^3 or −2x-2x or 77.Read more →Leading CoefficientThe coefficient of the highest-degree term in a polynomial written in standard form.Read more →Constant TermThe term with no variable factor — the value a0a_0 in a polynomial, equal to P(0)P(0).Read more →Degree (of a Polynomial)The highest exponent appearing on the variable in any term of the polynomial.Read more →MonomialA polynomial with exactly one term: a coefficient times a power of the variable, such as 5x25x^2.Read more →BinomialA polynomial with exactly two terms, such as x+3x + 3 or x2−4x^2 - 4.Read more →TrinomialA polynomial with exactly three terms, such as x2+5x+6x^2 + 5x + 6.Read more →Like TermsTerms that share the same variable raised to the same exponent, differing only in their coefficients.Read more →Root (of a Polynomial)A value rr such that P(r)=0P(r) = 0. Also called a zero or solution of the polynomial equation P(x)=0P(x) = 0.Read more →MultiplicityThe number of times a root rr appears as a factor (x−r)k(x - r)^k in the polynomial's factorization; kk is the multiplicity.Read more →FactoringWriting a polynomial as a product of two or more polynomials of lower degree.Read more →Greatest Common Factor (GCF)The largest expression — numerical and variable parts combined — that divides evenly into every term of a polynomial.Read more →Difference of Squaresa2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)Read more →Perfect Square Trinomiala2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2 and a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2.Read more →Sum and Difference of Cubesa3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2).Read more →Irreducible PolynomialA polynomial that cannot be factored into polynomials of lower degree over a given number system.Read more →End BehaviorThe direction the graph of a polynomial heads as x→+∞x \to +\infty and x→−∞x \to -\infty, determined by the leading term anxna_nx^n.Read more →Turning PointA point where a polynomial's graph changes from increasing to decreasing or vice versa; a polynomial of degree nn has at most n−1n - 1 turning points.Read more →Remainder TheoremWhen P(x)P(x) is divided by (x−c)(x - c), the remainder equals P(c)P(c).Read more →Factor Theorem(x−c)(x - c) is a factor of P(x)P(x) if and only if P(c)=0P(c) = 0.Read more →Rational Root TheoremIf pq\frac{p}{q} (in lowest terms) is a rational root of anxn+⋯+a0a_nx^n + \cdots + a_0, then pp divides a0a_0 and qq divides ana_n.Read more →Descartes' Rule of SignsThe number of positive real roots of P(x)P(x) equals the number of sign changes in its coefficients, or less by an even number. For negative roots,…Read more →Fundamental Theorem of AlgebraEvery polynomial of degree n≥1n \geq 1 with complex coefficients has exactly nn roots in C\mathbb{C}, counted with multiplicity.Read more →Vieta's FormulasFor P(x)=anxn+⋯+a0P(x) = a_n x^n + \cdots + a_0 with roots r1,…,rnr_1, \ldots, r_n: r1+r2+⋯+rn=−an−1anr_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n} and…Read more →Synthetic DivisionA shorthand method for dividing a polynomial by a linear binomial (x−c)(x - c) using only coefficients and the value cc.Read more →Polynomial Long DivisionA systematic procedure for dividing one polynomial by another, producing a quotient and a remainder: P(x)=D(x)⋅Q(x)+R(x)P(x) = D(x) \cdot Q(x) + R(x).Read more →
PowerAn expression ana^n consisting of a base aa raised to an exponent nn.Read more →Base (of a Power)The number aa in ana^n — the quantity being raised to a power.Read more →ExponentThe number nn in ana^n that controls how the base is used — counting repetitions for natural exponents, and generalizing through zero, negative,…Read more →Natural ExponentA positive integer exponent: an=a⋅a⋅a⋯a⏟n timesa^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}} for n≥1n \geq 1.Read more →Zero Exponenta0=1a^0 = 1 for any a≠0a \neq 0.Read more →Negative Exponenta−n=1ana^{-n} = \frac{1}{a^n} for a≠0a \neq 0.Read more →Irrational ExponentAn exponent that cannot be expressed as a fraction, such as π\pi or 2\sqrt{2}. The value axa^x is defined as the limit of ara^r as rational rr…Read more →Product Rule (Exponents)am⋅an=am+na^m \cdot a^n = a^{m+n} — same base, add exponents.Read more →Quotient Rule (Exponents)aman=am−n\frac{a^m}{a^n} = a^{m-n} — same base, subtract exponents.Read more →Power of a Power(am)n=amn(a^m)^n = a^{mn} — raise a power to a power, multiply exponents.Read more →Power of a Product(ab)n=an⋅bn(ab)^n = a^n \cdot b^n — distribute the exponent across multiplication.Read more →Power of a Quotient(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} — distribute the exponent across division, b≠0b \neq 0.Read more →Exponential EquationAn equation in which the variable appears in the exponent: af(x)=ba^{f(x)} = b.Read more →Exponential InequalityAn inequality involving an exponential expression, where the base determines whether inequality direction is preserved or reversed.Read more →Exponential FunctionA function of the form f(x)=axf(x) = a^x with fixed base a>0a > 0, a≠1a \neq 1, and variable exponent xx.Read more →
InequalityA statement comparing two expressions using <<, >>, ≤\leq, or ≥\geq, whose solution is typically an interval or union of intervals.Read more →Interval NotationA compact notation for solution sets: parentheses ( )(\,) for excluded endpoints, brackets [ ][\,] for included endpoints, with ∞\infty always…Read more →Compound InequalityTwo inequalities joined by AND (conjunction, intersection) or OR (disjunction, union), producing a combined solution set.Read more →Sign AnalysisA method for solving non-linear inequalities by finding all roots and undefined points, partitioning the number line into intervals, and determining…Read more →Critical PointA value where the expression equals zero (numerator zero) or is undefined (denominator zero), used to partition the number line for sign analysis.Read more →Linear InequalityAn inequality of the form ax+b<0ax + b < 0 (or >>, ≤\leq, ≥\geq) with a≠0a \neq 0; solution is always a single ray.Read more →Quadratic InequalityAn inequality of the form ax2+bx+c>0ax^2 + bx + c > 0 (or <<, ≤\leq, ≥\geq) with a≠0a \neq 0; solved via the discriminant and sign analysis.Read more →Polynomial InequalityAn inequality P(x)>0P(x) > 0 (or <<, ≤\leq, ≥\geq) where P(x)P(x) has degree n≥3n \geq 3; solved by sign analysis with attention to root multiplicity.Read more →Rational InequalityAn inequality involving a rational expression P(x)Q(x)\frac{P(x)}{Q(x)}; solved by sign analysis using both numerator zeros and denominator zeros as…Read more →Absolute Value Inequality∣f(x)∣<k|f(x)| < k converts to −k<f(x)<k-k < f(x) < k (conjunction); ∣f(x)∣>k|f(x)| > k converts to f(x)<−kf(x) < -k or f(x)>kf(x) > k (disjunction).Read more →
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Equations

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Complete guide to equations: definitions, solution sets, equation types (conditional, identity, contradiction), equivalent equations, and solving linear, quadratic, polynomial, rational, and absolute value equations.
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Algebraic Identities

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Master algebraic identities: squares and cubes of binomials, binomial theorem, trinomial expansions, difference of squares, sum and difference of cubes, higher-power factorizations, and special identities.

Algebraic identities are equations that hold true for every value of their variables — not just for specific solutions, but universally. The expression (a+b)2 = a2 + 2ab + b2 isn't a problem to solve; it's a structural fact about how squaring distributes over addition. When you treat an identity as a transformation rule rather than an equation to be balanced, algebra opens up.

Identities serve three roles. First, they compress: rather than expanding (a+b)(a−b) every time, you recognize the difference of squares and write a2−b2 instantly. Second, they reveal structure: factoring x3−y3 as (x−y)(x2+xy+y2) exposes a pattern that recurs across number theory and abstract algebra. Third, they enable substitution — trigonometric, logarithmic, and exponential identities let you rewrite expressions into forms where new techniques apply.

The basic binomial identities — squares, cubes, differences, and sums of cubes — form the backbone. Mastering them isn't memorization; it's pattern recognition. Once (a+b)2 lives in your head as a geometric square dissected into four pieces (a2, ab, ab, b2), you no longer expand it mechanically. You see it.

Below, each identity is presented with its algebraic form, geometric intuition where applicable, and the contexts where it does its heaviest work.

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Inequalities in Algebra

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Master solving inequalities: interval notation, sign analysis, compound inequalities, and methods for linear, quadratic, polynomial, rational, and absolute value inequalities with number line graphs.
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Logarithms

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Learn logarithms from definition to application: base and argument rules, inverse identities, common and natural logs, properties, rules, equations, and graphs.
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Polynomials

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Master polynomials: definition, degree, terms, standard form, evaluation, operations, factoring, roots, graphing, and key theorems including Remainder and Factor Theorems.
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Powers and Exponents

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Learn powers and exponents: natural, zero, negative, rational, and irrational exponents. Master exponent laws, exponential equations, inequalities, and functions.
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Roots and Radicals

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Learn roots and radicals: square roots, cube roots, principal roots, radical notation, rational exponents, simplification rules, radical equations, and functions.
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Sequences in Mathematics

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Learn about sequences: arithmetic, geometric, harmonic, Fibonacci, triangular, square, and prime numbers. Definitions, formulas, and recursive vs explicit forms.
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Algebra Calculators and Solvers

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Step-by-step calculators for algebra: equation solvers (linear, quadratic, polynomial, rational, radical, exponential, absolute value, literal), inequality solvers across the same families, and integer-sequence explorers (triangular, square, tetrahedral, Fibonacci, primes, arithmetic, geometric).
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Logarithmic Inequality Solver

Free, step-by-step solver for logarithmic inequalities. Handles domain restrictions, base-direction logic, and converts to exponential form to solve.

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Absolute Value Inequality Solver

Free, step-by-step solver for absolute value inequalities. Splits into compound or union form, handles edge cases, returns interval notation.

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x3 −2x2 −x +2 (x−1) (x+1) (x−2)
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Polynomial Calculator

Add, subtract, multiply, divide, factor, find zeros, or simplify polynomials in one tool. Eight operations, keypad or slots input, and full step-by-step solutions with method notes and common pitfalls for every operation.

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