What are algebra identities?

Algebra identities are equalities between two expressions that hold for all values of their variables. The core set includes the difference of squares, the perfect square trinomials, the sum and difference of cubes, and the cube of a binomial. They are the algebraic building blocks for factoring, simplifying, and expanding polynomial expressions.

What is the difference of squares identity?

The difference of squares says a squared minus b squared equals (a plus b) times (a minus b). It applies whenever a polynomial can be written as one square subtracted from another, even when a and b are themselves compound expressions. The identity is one of the fastest factoring shortcuts in algebra.

What is the perfect square trinomial identity?

The perfect square trinomial identities are (a plus b) squared equals a squared plus 2ab plus b squared, and (a minus b) squared equals a squared minus 2ab plus b squared. The middle term is always twice the product of the two parts. Recognizing this pattern lets you factor a trinomial back into a binomial squared at sight.

What is the difference between the sum and difference of cubes?

Sum of cubes: a cubed plus b cubed equals (a plus b) times (a squared minus ab plus b squared). Difference of cubes: a cubed minus b cubed equals (a minus b) times (a squared plus ab plus b squared). The SOAP mnemonic helps: Same sign on the binomial, Opposite sign in the middle of the trinomial, Always Positive on the last term.

How do I memorize algebra identities?

Group them by family: difference-of-powers (squares and cubes), perfect-power expansions (square and cube of a binomial), and multi-term expansions like (a plus b plus c) squared. Notice the structural patterns: the middle term in a perfect square is always twice the product of the parts; cubes split into a binomial times a trinomial. Drill the drag puzzle on this page to lock the pairings in.

Algebraic Identities

Reference table of algebraic identities — equations that hold for all values in their domain. Try puzzle mode to drill, or browse the full algebra section →

Algebraic Identities Tool

Search by LHS, by RHS, by name — or pick a family.

View

Rows view: dense, print-friendly listing. Click any row to expand.

Or skip to the quiz →

Name	Left-hand side		Right-hand side	Family
Distributive Identity	$a(b + c)$	=	$ab + ac$	Distributive
FOIL Identity	$(a + b)(c + d)$	=	$ac + ad + bc + bd$	Distributive
Square of a Sum Identity	$(a + b)^2$	=	$a^2 + 2ab + b^2$	Squares & Cubes
Square of a Difference Identity	$(a - b)^2$	=	$a^2 - 2ab + b^2$	Squares & Cubes
Triple Square Identity	$(a + b + c)^2$	=	$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$	Squares & Cubes
Cube of a Sum Identity	$(a + b)^3$	=	$a^3 + 3a^2 b + 3ab^2 + b^3$	Squares & Cubes
Cube of a Difference Identity	$(a - b)^3$	=	$a^3 - 3a^2 b + 3ab^2 - b^3$	Squares & Cubes
Difference of Squares Identity	$a^2 - b^2$	=	$(a + b)(a - b)$	Factoring
Sum of Cubes Identity	$a^3 + b^3$	=	$(a + b)(a^2 - ab + b^2)$	Factoring
Difference of Cubes Identity	$a^3 - b^3$	=	$(a - b)(a^2 + ab + b^2)$	Factoring
Difference of Even Powers Identity	$x^{2n} - a^{2n}$	=	$(x^n - a^n)(x^n + a^n)$	Factoring
Difference of Powers Identity (odd n)	$x^n - a^n$	=	$(x - a)(x^{n-1} + a x^{n-2} + \cdots + a^{n-1})$	Factoring
Sum of Powers Identity (odd n)	$x^n + a^n$	=	$(x + a)(x^{n-1} - a x^{n-2} + \cdots + a^{n-1})$	Factoring
Trinomial Factoring Identity	$x^2 + (a + b)x + ab$	=	$(x + a)(x + b)$	Factoring
Binomial Theorem Identity	$(x + y)^n$	=	$\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$	Binomial Expansion
Binomial Coefficient Identity	$\binom{n}{k}$	=	$\dfrac{n!}{k!\,(n-k)!}$	Binomial Expansion
Pascal's Identity	$\binom{n}{k} + \binom{n}{k+1}$	=	$\binom{n+1}{k+1}$	Binomial Expansion
Binomial Symmetry Identity	$\binom{n}{n-k}$	=	$\binom{n}{k}$	Binomial Expansion
Product Rule Identity (Exponents)	$a^m \cdot a^n$	=	$a^{m+n}$	Exponent Laws
Quotient Rule Identity (Exponents)	$\dfrac{a^m}{a^n}$	=	$a^{m-n}$	Exponent Laws
Power of a Power Identity	$(a^m)^n$	=	$a^{mn}$	Exponent Laws
Power of a Product Identity	$(ab)^n$	=	$a^n \, b^n$	Exponent Laws
Power of a Quotient Identity	$\left(\dfrac{a}{b}\right)^n$	=	$\dfrac{a^n}{b^n}$	Exponent Laws
Zero Exponent Identity	$a^0$	=	$1$	Exponent Laws
Negative Exponent Identity	$a^{-n}$	=	$\dfrac{1}{a^n}$	Exponent Laws
Negative Exponent Flip Identity	$\left(\dfrac{a}{b}\right)^{-n}$	=	$\left(\dfrac{b}{a}\right)^n$	Exponent Laws
Product Rule Identity (Logarithms)	$\log_a(xy)$	=	$\log_a(x) + \log_a(y)$	Logarithm Identities
Quotient Rule Identity (Logarithms)	$\log_a\!\left(\dfrac{x}{y}\right)$	=	$\log_a(x) - \log_a(y)$	Logarithm Identities
Power Rule Identity (Logarithms)	$\log_a(x^n)$	=	$n \, \log_a(x)$	Logarithm Identities
Change of Base Identity	$\log_a(x)$	=	$\dfrac{\log_b(x)}{\log_b(a)}$	Logarithm Identities
Logarithm of the Base Identity	$\log_a(a)$	=	$1$	Logarithm Identities
Logarithm of One Identity	$\log_a(1)$	=	$0$	Logarithm Identities
Logarithm of an Exponential Identity	$\log_a(a^x)$	=	$x$	Logarithm Identities
Exponential of a Logarithm Identity	$a^{\log_a(x)}$	=	$x$	Logarithm Identities
Radical to Exponent Identity	$\sqrt[n]{a}$	=	$a^{1/n}$	Radical Rules
Product Rule Identity (Radicals)	$\sqrt[n]{ab}$	=	$\sqrt[n]{a} \cdot \sqrt[n]{b}$	Radical Rules
Quotient Rule Identity (Radicals)	$\sqrt[n]{\dfrac{a}{b}}$	=	$\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$	Radical Rules
Power Rule Identity (Radicals)	$\sqrt[n]{a^m}$	=	$a^{m/n}$	Radical Rules
Nested Radicals Identity	$\sqrt[m]{\sqrt[n]{a}}$	=	$\sqrt[mn]{a}$	Radical Rules
Absolute Value Product Identity	$\|ab\|$	=	$\|a\| \, \|b\|$	Absolute Value
Absolute Value Quotient Identity	$\left\|\dfrac{a}{b}\right\|$	=	$\dfrac{\|a\|}{\|b\|}$	Absolute Value
Absolute Value Negation Identity	$\|-a\|$	=	$\|a\|$	Absolute Value
Absolute Value Square Identity	$\|a\|^2$	=	$a^2$	Absolute Value
Common Denominator Addition Identity	$\dfrac{a}{c} + \dfrac{b}{c}$	=	$\dfrac{a + b}{c}$	Fraction Operations
Different Denominator Addition Identity	$\dfrac{a}{b} + \dfrac{c}{d}$	=	$\dfrac{ad + bc}{bd}$	Fraction Operations
Fraction Multiplication Identity	$\dfrac{a}{b} \cdot \dfrac{c}{d}$	=	$\dfrac{ac}{bd}$	Fraction Operations
Fraction Division Identity	$\dfrac{a}{b} \div \dfrac{c}{d}$	=	$\dfrac{ad}{bc}$	Fraction Operations
Quadratic Formula Identity	$x_{1,2}$	=	$\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Polynomial Identities
Discriminant Identity	$\Delta$	=	$b^2 - 4ac$	Polynomial Identities
Completing the Square Identity	$x^2 + bx$	=	$\left(x + \dfrac{b}{2}\right)^2 - \dfrac{b^2}{4}$	Polynomial Identities
Vieta Sum Identity	$x_1 + x_2$	=	$-\dfrac{b}{a}$	Polynomial Identities
Vieta Product Identity	$x_1 \, x_2$	=	$\dfrac{c}{a}$	Polynomial Identities
Remainder Theorem Identity	$P(x)$	=	$(x - c) \, Q(x) + P(c)$	Polynomial Identities

Families of identities

Click a family to highlight its entries in the table above.

a(b+c)

Distributive

How multiplication distributes over addition.

2 matchesClick to highlight

(a+b)²

Squares & Cubes

Expansions of binomial and trinomial powers.

5 matchesClick to highlight

a²−b²

Factoring

Factoring patterns for differences and sums.

7 matchesClick to highlight

(x+y)ⁿ

Binomial Expansion

The binomial theorem and Pascal's triangle.

4 matchesClick to highlight

aⁿ

Exponent Laws

Rules for combining and simplifying powers.

8 matchesClick to highlight

log

Logarithm Identities

Rules for combining and converting logarithms.

8 matchesClick to highlight

√

Radical Rules

Rules for $n$ -th roots and rational exponents.

5 matchesClick to highlight

|a|

Absolute Value

Identities involving the absolute value function.

4 matchesClick to highlight

a/b

Fraction Operations

Arithmetic of rational expressions.

4 matchesClick to highlight

P(x)

Polynomial Identities

Identities used to solve and analyze polynomials.

6 matchesClick to highlight

Field axioms

The structural rules governing arithmetic on real numbers.

⊕

Commutativity

Addition and multiplication of real numbers are commutative — order does not affect the result.

a + b = b + a, \quad ab = ba

∘

Associativity

Addition and multiplication are associative — grouping does not affect the result.

(a + b) + c = a + (b + c), \quad (ab)c = a(bc)

⊗

Distributivity

Multiplication distributes over addition. This is the structural principle linking the two operations.

a(b + c) = ab + ac

∅

Identity elements

$0$ is the additive identity ( $a + 0 = a$ ); $1$ is the multiplicative identity ( $a \cdot 1 = a$ ). These leave their operands unchanged.

7 + 0 = 7, \quad 7 \cdot 1 = 7

−

Inverse elements

Every $a$ has an additive inverse $-a$ such that $a + (-a) = 0$ . Every nonzero $a$ has a multiplicative inverse $1/a$ such that $a \cdot (1/a) = 1$ .

5 + (-5) = 0, \quad 5 \cdot \dfrac{1}{5} = 1

Key Terms

What This Table Covers

Using the Table

Drill With the Puzzle and Quiz

Distributive and Expansion Identities

Factoring Patterns and the Binomial Theorem

Exponent Laws

Logarithm Identities

Radicals, Absolute Value, and Fractions

Polynomial Identities

Common Mistakes

Related Concepts

Key Terms

Identity — An equation that holds for every value of its variables, not just specific ones. Every row in the table is an identity.

Left-hand side (LHS) — The expression on the left of the equals sign. Usually the form you start with.

Right-hand side (RHS) — The expression on the right. Usually the form you want, whether that means a factored form, an expanded form, or a simpler equivalent.

Family — The structural category an identity belongs to: distributive, expansion, factoring, binomial, exponent, logarithm, radical, absolute value, fraction, or polynomial. Each family corresponds to a colored badge in the table and a card in the categories grid.

Filter — Clicking a family card highlights only that family's rows. Search and filter both highlight; only one is active at a time.

Tip — The short explanatory note attached to each identity card or row, explaining when the identity applies or how to remember it.

What This Table Covers

The table catalogues 53 algebra identities across 10 structural families. The full inventory:

Distributive identities (2) — the rule $a(b+c) = ab + ac$ and FOIL.
Squares and cubes expansions (5) — $(a \pm b)^2$ , $(a \pm b)^3$ , and $(a+b+c)^2$ .
Factoring patterns (7) — difference of squares, sum and difference of cubes, higher-power differences, and the trinomial pattern.
Binomial expansion (4) — the binomial theorem, the coefficient $\binom{n}{k}$ , Pascal's identity, and binomial symmetry.
Exponent laws (8) — product, quotient, power-of-power, distribution over products and quotients, zero, negative, and reciprocal-flip rules.
Logarithm identities (8) — the three core rules, change of base, and four inverse-function identities.
Radical rules (5) — the radical-to-exponent bridge and the product, quotient, power, and nesting rules.
Absolute value (4) — how $|\cdot|$ interacts with multiplication, division, negation, and squaring.
Fraction operations (4) — same-denominator addition, different-denominator addition, multiplication, and division.
Polynomial identities (6) — quadratic formula, discriminant, completing the square, Vieta's formulas, and the remainder theorem.

Each row shows the LHS, the RHS, the family, a one-line tip, and a link to the formula entry where available.

Learn More

Using the Table

The table supports three views, toggled by the buttons above the grid.

Cards — one card per identity, with the LHS and RHS stacked. Click any card to expand its details panel inline.
Rows — dense, print-friendly tabular layout. Same click-to-expand behavior. Default view on load.
Puzzle — drag-and-drop drill mode. Covered separately in the puzzle and quiz section.

The search tool above the table accepts four search modes: by name, by LHS, by RHS, and by category. Type in the field and matches highlight live. The Go → button after a successful search scrolls the result into view.

The action bar to the right of the view toggle has three buttons:

Print — opens the browser print dialog with the current view.
Copy CSV — copies all entries to the clipboard as CSV with columns Name, LHS, RHS, Category, Tip, Link.
Download .csv — downloads the same CSV as a file.

The categories grid acts as a one-click filter. Click any family card to highlight just its entries; click again or use Clear highlight to reset.

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Drill With the Puzzle and Quiz

Two practice modes are built into the page.

Puzzle mode is the third view above the table. Switch to it and the table reorganizes: each LHS becomes an empty slot, each RHS becomes a draggable tile. Drop the right tile into the right slot. Tiles that don't match snap back. Each tile has a hint button for a nudge and a see-why link to the formula page once placed correctly.

Quiz mode sits in its own section at the bottom of the page. The widget rotates three question types at random:

• Given an LHS, pick the correct RHS from four options.
• Given an RHS, pick the correct LHS from four options.
• Given the full identity, pick the family it belongs to.

Score is tracked across the session. The widget keeps a 30-question history so you can review missed questions. Reset clears the score and starts fresh.

Use the puzzle for active recall and the quiz for spaced rotation. Both pull from the same 53-item table.

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Distributive and Expansion Identities

The distributive family has two entries that anchor everything else. The distributive identity

a(b+c) = ab + ac

is the structural rule linking addition and multiplication. FOIL —

(a+b)(c+d) = ac + ad + bc + bd

— applies the distributive identity twice to multiply two binomials.

The squares and cubes family collects five expansions:

$(a+b)^2 = a^2 + 2ab + b^2$ — the perfect-square trinomial.
$(a-b)^2 = a^2 - 2ab + b^2$ — same coefficients, signed cross term.
$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ — the trinomial square; each variable squared, plus twice every unordered pair product.
$(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$ — coefficients from the third row of Pascal's triangle: 1, 3, 3, 1.
$(a-b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3$ — same coefficients, alternating signs.

Memorize the perfect-square and perfect-cube patterns first. They are the entry point to factoring and the basis for completing the square.

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Factoring Patterns and the Binomial Theorem

The factoring family is the most-used set in algebra. Seven identities:

Difference of squares: $a^2 - b^2 = (a+b)(a-b)$ . The most-applied factoring shortcut.
Sum of cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ .
Difference of cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ . SOAP mnemonic: Same sign, Opposite sign, Always Positive.
Difference of even powers: $x^{2n} - a^{2n} = (x^n - a^n)(x^n + a^n)$ .
Difference of powers (any $n$ ): $x^n - a^n$ always has $(x-a)$ as a factor.
Sum of powers (odd $n$ only): $x^n + a^n$ has $(x+a)$ as a factor.
Trinomial factoring: $x^2 + (a+b)x + ab = (x+a)(x+b)$ .

Sum of squares does not factor over the reals.

a^2 + b^2

is irreducible.

The binomial expansion family generalizes

(a+b)^n

to arbitrary powers. The binomial theorem

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

expands any non-negative integer power. The binomial coefficient

\binom{n}{k} = n! / (k!(n-k)!)

counts subsets. Pascal's identity

\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}

generates the triangle row by row. Binomial symmetry

\binom{n}{k} = \binom{n}{n-k}

reflects the include-or-exclude duality.

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Exponent Laws

The exponent family has eight identities covering every legal operation on powers with a common base or exponent.

Product rule: $a^m \cdot a^n = a^{m+n}$ . Same base → add exponents.
Quotient rule: $a^m / a^n = a^{m-n}$ . Same base → subtract exponents.
Power of a power: $(a^m)^n = a^{mn}$ . Multiply exponents.
Power of a product: $(ab)^n = a^n b^n$ . Distributes over multiplication.
Power of a quotient: $(a/b)^n = a^n / b^n$ . Distributes over division.
Zero exponent: $a^0 = 1$ for any nonzero $a$ . The expression $0^0$ is indeterminate.
Negative exponent: $a^{-n} = 1/a^n$ . Means reciprocal, not a negative result.
Negative exponent flip: $(a/b)^{-n} = (b/a)^n$ . Invert and switch sign on the exponent.

These eight rules are the foundation for logarithm rules and radical rules. Both families re-derive immediately from exponents via the inverse and rational-exponent bridges.

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Logarithm Identities

The logarithm family has eight identities that mirror the exponent rules because logs are inverse exponentials.

Three core transformation rules:

Product: $\log_a(xy) = \log_a(x) + \log_a(y)$ . Multiplication inside becomes addition outside.
Quotient: $\log_a(x/y) = \log_a(x) - \log_a(y)$ . Division becomes subtraction.
Power: $\log_a(x^n) = n \log_a(x)$ . Exponent comes out as a coefficient.

Change of base:

\log_a(x) = \log_b(x) / \log_b(a)

. Converts to any chosen base, including the natural log

\ln

or common log

\log_{10}

for calculator evaluation.

Four trivial identities that keep showing up:

$\log_a(a) = 1$ (the log of the base).
$\log_a(1) = 0$ (the log of one is always zero, regardless of base).
$\log_a(a^x) = x$ (log undoes exponential).
$a^{\log_a(x)} = x$ (exponential undoes log, for $x > 0$ ).

The last pair are inverse-function identities. They are the reason logs are the canonical solving tool for

a^x = b

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Radicals, Absolute Value, and Fractions

Three smaller families round out the structural identities.

[Radical rules](!/tables/algebra-identities#radical) (5 identities) reduce to exponent rules via the bridge

\sqrt[n]{a} = a^{1/n}

. The product rule

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

, the quotient rule, and the power rule follow immediately. Nested radicals collapse via

\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}

. Even-index radicals require non-negative operands to stay real.

[Absolute value identities](!/tables/algebra-identities#absoluteValue) (4 identities) preserve multiplicative structure but not additive:

$|ab| = |a| \, |b|$ .
$|a/b| = |a| / |b|$ .
$|-a| = |a|$ .
$|a|^2 = a^2$ .

Note:

|a + b| \neq |a| + |b|

in general. Only the triangle inequality

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

holds.

[Fraction operations](!/tables/algebra-identities#fraction) (4 identities) cover the four arithmetic combinations: same-denominator addition (just add numerators), different-denominator addition (cross-multiply to

bd

), multiplication (numerators times numerators), and division (multiply by reciprocal).

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Polynomial Identities

The polynomial family collects six identities used to solve and analyze polynomial equations.

Quadratic formula: the roots of

ax^2 + bx + c = 0

are

x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)

, written directly from the coefficients.

Discriminant

\Delta = b^2 - 4ac

determines the root structure without solving:

$\Delta > 0$ : two distinct real roots.
$\Delta = 0$ : one repeated real root.
$\Delta < 0$ : two complex conjugate roots.

Completing the square:

x^2 + bx = (x + b/2)^2 - b^2/4

. The algebraic move behind the quadratic formula and the vertex form of a parabola.

Vieta's formulas extract sum and product of roots from coefficients:

$x_1 + x_2 = -b/a$ (sum of roots).
$x_1 x_2 = c/a$ (product of roots).

Useful when only the sum or product is needed — no need to solve the equation.

Remainder theorem:

P(x) = (x - c) Q(x) + P(c)

. Dividing a polynomial by

(x - c)

leaves remainder

P(c)

— no long division required. The basis of synthetic division and the factor theorem (

P(c) = 0 \iff (x-c)

is a factor).

Learn More

Common Mistakes

Six errors that account for most algebraic slip-ups:

$(a+b)^2 \neq a^2 + b^2$. The middle term $2ab$ is real. Dropping it is the single most common algebra error. The full identity is $(a+b)^2 = a^2 + 2ab + b^2$ from the expansion family.

$a^2 + b^2$ does not factor over the reals. Only the difference of squares factors. The sum of squares is irreducible (it factors over the complex numbers as $(a+bi)(a-bi)$ , but not over the reals).

$\log(x+y) \neq \log(x) + \log(y)$. The log product rule applies to multiplication inside, not addition. Log of a sum has no clean expansion.

$a^{m+n} \neq a^m + a^n$. The product rule for exponents governs multiplication, not addition. Adding exponents is the result of multiplying powers, never of adding them.

A negative exponent does not negate the value. $a^{-n} = 1/a^n$ , a reciprocal. $2^{-3} = 1/8$ , not $-8$ .

Negative discriminant does not mean no roots. $\Delta < 0$ in the quadratic family means two complex conjugate roots, not zero roots.

Related Concepts

[Field axioms](!/tables/algebra-identities#sec-properties) — the structural rules (commutativity, associativity, distributivity, identity elements, inverse elements) that justify every identity in the table. Listed at the bottom of this page below the categories grid. Worth a glance for the underlying logic, especially when an identity seems arbitrary.

Algebra formulas — the formula index where each identity in this table has a derivation page. Card details panels link directly to the relevant entry.

Polynomial factoring — the practical application of the factoring family. See the polynomial factoring topic for step-by-step worked examples.

Quadratic equations — uses the polynomial family heavily. Quadratic formula, discriminant analysis, Vieta's formulas, and completing the square all derive from there.

Exponent rules and logarithm rules — the topic pages that develop the exponent and logarithm families with examples and exercises.

Binomial theorem — the combinatorial backbone of the binomial expansion family. See the binomial theorem topic for proofs and applications to probability and series.

Learn More