What are the essential algebraic identities?

They are the square of a sum, the square of a difference, the difference of squares, the cube of a sum, the cube of a difference, the sum of two cubes, and the difference of two cubes. These seven special products are the foundation of introductory algebra.

What are algebraic identities used for?

They let you expand expressions quickly, factor polynomials, simplify expressions, and solve equations without multiplying every term by hand. Three of them (difference of squares, sum of cubes, difference of cubes) are especially useful for factoring.

How do I tell (a − b)³ apart from a³ − b³?

They are two different formulas. (a − b)³ = a³ − 3a²b + 3ab² − b³ expands to four terms, while a³ − b³ = (a − b)(a² + ab + b²) is a difference of two cubes written as a product of two factors.

Why does the sum of two cubes contain a² − ab + b²?

Because a³ + b³ = (a + b)(a² − ab + b²): the middle term ab always takes the opposite sign of the sign in the linear factor. With a + b, the quadratic factor uses a minus sign in the middle.

Algebraic Identities: The Essential Formulas with Examples

The essential algebraic identities are the backbone of all introductory algebra, and they follow you for years afterward — from expanding and simplifying expressions to factoring polynomials and solving equations. This article collects every formula together with expansion examples, factoring examples, and quick tricks to remember them.

The table of essential identities

#	Name	Formula
1	Square of a sum	$(a + b)^{2} = a^{2} + 2 ab + b^{2}$
2	Square of a difference	$(a - b)^{2} = a^{2} - 2 ab + b^{2}$
3	Difference of squares	$a^{2} - b^{2} = (a - b) (a + b)$
4	Cube of a sum	$(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$
5	Cube of a difference	$(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$
6	Sum of two cubes	$a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$
7	Difference of two cubes	$a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

Memorizing the table is the first step, but it matters far more to understand why each formula is true. Let's walk through them group by group.

The square identities

Square of a sum and square of a difference

The first two formulas differ only in the sign of the middle term:

$(a + b)^{2} = a^{2} + 2 ab + b^{2}$

$(a - b)^{2} = a^{2} - 2 ab + b^{2}$

Both read the same way: "square the first term, plus (or minus) twice the product of the two terms, plus the square of the second term."

Expansion example. Expand $(x + 3)^{2}$ and $(2 x - 5)^{2}$ .

$(x + 3)^{2} = x^{2} + 2 \cdot x \cdot 3 + 3^{2} = x^{2} + 6 x + 9$

$(2 x - 5)^{2} = (2 x)^{2} - 2 \cdot 2 x \cdot 5 + 5^{2} = 4 x^{2} - 20 x + 25$

Difference of squares

This is the identity used most often for factoring:

$a^{2} - b^{2} = (a - b) (a + b)$

Factoring example. Factor $9 x^{2} - 16$ .

Write $9 x^{2} = (3 x)^{2}$ and $16 = 4^{2}$ , so:

$9 x^{2} - 16 = (3 x)^{2} - 4^{2} = (3 x - 4) (3 x + 4)$

The same identity speeds up mental arithmetic: $101 \cdot 99 = (100 + 1) (100 - 1) = 10 0^{2} - 1^{2} = 9999$ .

The cube identities

Cube of a sum and cube of a difference

$(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

$(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$

The difference: for $(a - b)^{3}$ the signs alternate in the order $+, -, +, -$ . The coefficients $1, 3, 3, 1$ are always the same.

Expansion example. Expand $(x + 2)^{3}$ .

$(x + 2)^{3} = x^{3} + 3 \cdot x^{2} \cdot 2 + 3 \cdot x \cdot 2^{2} + 2^{3} = x^{3} + 6 x^{2} + 12 x + 8$

Sum and difference of two cubes

The last two formulas rewrite a sum (or difference) of cubes as a product:

$a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$

$a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

A trick for the signs: the linear factor keeps the exact sign of the left-hand side ( $a + b$ or $a - b$ ), while the middle term $ab$ in the quadratic factor takes the opposite sign. The $a^{2}$ and $b^{2}$ terms are always added.

Factoring example. Factor $x^{3} + 8$ and $27 x^{3} - 1$ .

$x^{3} + 8 = x^{3} + 2^{3} = (x + 2) (x^{2} - 2 x + 4)$

$27 x^{3} - 1 = (3 x)^{3} - 1^{3} = (3 x - 1) (9 x^{2} + 3 x + 1)$

Tips to remember

The cube coefficients are $1, 3, 3, 1$ — exactly the fourth row of Pascal's triangle, which is easy to recall.
Signs in $(a - b)^{n}$ : any term with an odd power of $b$ (that is, the term with $b$ or with $b^{3}$ ) carries a minus sign.
Sum/difference of cubes: the quadratic factor is the "incomplete square" — incomplete because it is missing the coefficient $2$ compared with $(a \pm b)^{2}$ ; and the middle $ab$ takes the opposite sign of the first factor.
Identities 3, 6, and 7 always produce a product, so use them for factoring; the rest are used for expanding.

Common mistakes

Forgetting the middle term $2 ab$ : writing $(a + b)^{2} = a^{2} + b^{2}$ is wrong. There is always a $2 ab$ .
Confusing $(a - b)^{3}$ with $a^{3} - b^{3}$ : one expands to four terms, the other is a product of two factors — they are completely different.
Wrong sign on $ab$ in the sum/difference of cubes: for $a^{3} + b^{3}$ the quadratic factor is $a^{2} - ab + b^{2}$ (minus), while for $a^{3} - b^{3}$ it is $a^{2} + ab + b^{2}$ (plus) — remember the sign is opposite.
Swapping $a^{2} b$ and $a b^{2}$ : in the cube expansions, the term next to $a^{3}$ contains $a^{2} b$ , and the term next to $b^{3}$ contains $a b^{2}$ .

Master these identities and you'll expand and factor polynomials much faster, with far fewer sign slips. Try entering the expansion and factoring examples above into the calculator to check each step of your own work.