Algebraic Identities

Definition

An identity is an equality that holds true regardless of the values chosen for its variables.

For example, the identity $(x+y)^2 = x^2 + 2xy + y^2$ is always true regardless of the values of $x$ and $y$.

Technique

Since identities are true for all valid values of its variables, one side of the equality can be swapped for the other. For example, we can replace any instance of $(x+y)^2$ with $x^2 + 2xy + y^2$ and vice versa because $(x+y)^2 = x^2 + 2xy + y^2$ is an identity.

Clever use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.

These identities are product formulas that are basic examples of the binomial theorem. $\begin{aligned} (x+y)^2 &= x^2 + 2xy + y^2 \\ (x-y)^2 &= x^2 - 2xy + y^2 \\ (x+y)^3 &= x^3+3x^2y + 3xy^2 + y^3 \\ (x-y)^3 &= x^3-3x^2y + 3xy^2 - y^3 \\ (x+y)^4 &= x^4 + 4x^3y + 6x^2y^2+4xy^3 + y^4 \\ (x-y)^4 &= x^4 - 4x^3y + 6x^2y^2-4xy^3 + y^4 \end{aligned}$

These identities are factoring formulas. Their more general forms are listed on the factorization page.

$\begin{aligned} x^2 - y^2 &= (x+y)(x-y) \\ x^3 - y^3 &= (x-y)(x^2+xy+y^2) \\ x^3 + y^3 &= (x+y)(x^2-xy+y^2) \\ x^4 - y^4 &= (x^2-y^2)(x^2+y^2) \end{aligned}$

Application and Extensions

The identity $4(x+7)(2x-1)=Ax^2+Bx+C$ holds for all real values of $x$. What is $A+B+C$?

Multiplying out the left side of the identity we get $4(x+7)(2x-1)=8x^2+52x-28.$
This expression must be equal to the right hand side of the identity, therefore
$8x^2+52x-28=Ax^2+Bx+C,$
so $A=8$, $B=52$, and $C=-28$, which gives us $A+B+C=8+52-28=32$. $_\square$

If $A+B=8$ and $AB=13$, what is $A^3+B^3$?

While you could solve for $A$ and $B$, a more elegant solution exploits the identity $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
which can be rewritten as $x^3+y^3 = (x+y)^3-3xy(x+y).$
Substituting in $A$ and $B$ for $x$ and $y$ we get $\begin{aligned} A^3+B^3 &= (A+B)^3-3AB(A+B) \\ &= (8)^3-3(13)(8) \\ &= 512-312 \\ &= 200 \quad_\square \end{aligned}$