Fundamental Algebraic Identities

This lesson collects the standard algebraic identities used throughout school maths, first the ones involving squares and then the ones involving cubes.

Identities with squares

The square of a sum and the square of a difference of two terms:

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

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

The product of a sum and a difference gives a difference of squares:

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

Products of binomials

Multiplying two binomials that share a common term:

$(x+a)(x+b) = x^2 + (a+b)\,x + ab$

And the general product of two binomials:

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

Square of a three-term sum

For three terms, every pairwise product appears twice:

$(x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

Identities with cubes

The cube of a sum and the cube of a difference:

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

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

These can be rewritten in a compact form that is often handier:

$(a+b)^3 = a^3 + b^3 + 3ab(a+b)$

$(a-b)^3 = a^3 - b^3 - 3ab(a-b)$

Sum and difference of cubes

Factorising the sum of two cubes:

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

which rearranges to

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

Likewise the difference of two cubes:

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

which rearranges to

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

Identity for three cubes

The symmetric identity for three terms:

$a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)$

A useful special case follows when the three terms sum to zero:

$\text{if } a+b+c = 0 \text{, then } a^3 + b^3 + c^3 = 3abc$

Key takeaways

• The square identities give the sum, difference, and product forms, including the three-term square where each cross product is doubled.
• The cube of a binomial has both an expanded form and a compact form using $3ab(a\pm b)$.
• A sum or difference of cubes always factors out $(a+b)$ or $(a-b)$, and when $a+b+c=0$ the three cubes collapse to $3abc$.