An algebraic identity is an equality that is true for every allowed value of the variable.
Instead of solving for one special value, an identity describes a pattern that always works.
1. Identity vs. equation
An equation such as
\[
\begin{aligned}
x^2-9=0
\end{aligned}
\]
is true only for certain values of \(x\). In this case, it is true when \(x=3\) or \(x=-3\).
An identity is different. It is true for all values where the expressions are defined.
For example:
\[
\begin{aligned}
(x+3)^2
&=
x^2+6x+9
\end{aligned}
\]
is an identity because expanding \((x+3)^2\) always gives \(x^2+6x+9\).
2. How identities are verified
To verify a polynomial identity, expand both forms and compare the coefficients of equal powers.
If every coefficient matches, the identity is true.
\[
\begin{aligned}
(x+3)^2
&=
(x+3)(x+3)\\
&=
x^2+3x+3x+9\\
&=
x^2+6x+9.
\end{aligned}
\]
Since both sides reduce to the same polynomial, they are identically equal.
3. Perfect-square trinomial
The perfect-square identities are:
\[
\begin{aligned}
(a+b)^2
&=
a^2+2ab+b^2,\\
(a-b)^2
&=
a^2-2ab+b^2.
\end{aligned}
\]
A trinomial is a perfect square when the first and last terms are squares and the middle term is twice
the product of their square roots.
Example:
\[
\begin{aligned}
x^2+6x+9
&=
x^2+2\cdot x\cdot 3+3^2\\
&=
(x+3)^2.
\end{aligned}
\]
4. Difference of squares
The difference of squares identity is:
\[
\begin{aligned}
a^2-b^2
&=
(a-b)(a+b).
\end{aligned}
\]
Example:
\[
\begin{aligned}
x^2-16
&=
x^2-4^2\\
&=
(x-4)(x+4).
\end{aligned}
\]
This identity works because the middle terms cancel:
\[
\begin{aligned}
(a-b)(a+b)
&=
a^2+ab-ab-b^2\\
&=
a^2-b^2.
\end{aligned}
\]
5. Sum and difference of cubes
The cube identities are:
\[
\begin{aligned}
a^3+b^3
&=
(a+b)(a^2-ab+b^2),\\
a^3-b^3
&=
(a-b)(a^2+ab+b^2).
\end{aligned}
\]
Examples:
\[
\begin{aligned}
x^3+27
&=
x^3+3^3\\
&=
(x+3)(x^2-3x+9),
\end{aligned}
\]
\[
\begin{aligned}
x^3-8
&=
x^3-2^3\\
&=
(x-2)(x^2+2x+4).
\end{aligned}
\]
6. Cube of a binomial
The cube of a binomial expands as:
\[
\begin{aligned}
(a+b)^3
&=
a^3+3a^2b+3ab^2+b^3,\\
(a-b)^3
&=
a^3-3a^2b+3ab^2-b^3.
\end{aligned}
\]
Example:
\[
\begin{aligned}
(x+2)^3
&=
x^3+3x^2(2)+3x(2^2)+2^3\\
&=
x^3+6x^2+12x+8.
\end{aligned}
\]
7. Factorable quadratic pattern
Some quadratics match the product pattern:
\[
\begin{aligned}
x^2+(m+n)x+mn
&=
(x+m)(x+n).
\end{aligned}
\]
Example:
\[
\begin{aligned}
x^2+5x+6
&=
(x+2)(x+3),
\end{aligned}
\]
because \(2+3=5\) and \(2\cdot 3=6\).
8. Why coefficient comparison works
Two polynomials are identical when every coefficient of every matching power is the same.
For example:
\[
\begin{aligned}
x^2+6x+9
\quad\text{and}\quad
(x+3)^2
\end{aligned}
\]
both expand to:
\[
\begin{aligned}
x^2+6x+9.
\end{aligned}
\]
Therefore, their \(x^2\), \(x\), and constant coefficients all match.
9. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
10. Common mistakes
- Forgetting the middle term in \((a+b)^2\).
- Writing \((a+b)^2=a^2+b^2\), which is false.
- Using the wrong sign in sum/difference of cubes.
- Confusing \(a^2-b^2\) with \(a^2+b^2\). The sum of squares does not factor over the real numbers in the same way.
- Checking only one value of \(x\) and assuming the result is an identity.
- Not expanding fully before comparing expressions.
Key idea: expand, collect like terms, then compare the result with known algebraic patterns.
