Polynomial Identity

A polynomial identity is an equation between polynomials that holds for every value of the variable, such as (a + b)² = a² + 2ab + b² or the difference of squares. Below: how to spot the structure before expanding, prove an identity by matching coefficients, and the common expansion errors.

Tutor-style math help

Polynomial Identity: what to notice and how to work it

Polynomials skill

Polynomial problems reward structure. Before expanding, look for degree, leading term, common factors, and familiar products.

What to notice first

Put the polynomial in standard form when possible. The leading term tells end behavior, and factors reveal zeros.

Common student mistake

Do not cancel or combine unlike terms. \(x^2\), \(x\), and constants are different kinds of terms.

Key formulas and cues

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

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

\(P(c)=0\Rightarrow (x-c)\text{ is a factor}\)

A reliable path

Organize by degreeWrite terms from highest power to lowest power.
Look for structureTry GCF, special products, grouping, or division depending on the expression.
Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.

Worked examples

Combine like terms

Example: \(3x^2+5x-x^2+2x\)

Group x squared terms.
Group x terms.
Combine each group.

Answer: \(2x^2+7x\)

Factor a difference of squares

Example: \(x^2-25\)

Recognize a squared term minus a squared term.
Use a^2 – b^2.
Write conjugate factors.

Answer: \((x-5)(x+5)\)

Open pop-up practice

Try one before moving on

Try: Factor \(x^2+7x+12\).

Answer: \((x+3)(x+4)\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Polynomial identities are equations that hold true for all possible values of the variable. When solving problems with polynomial identities, identify the pattern to see if the form is simplified or factored form, and then apply the identity and solve.

A step-by-step guide to polynomial identity

Polynomial identity refers to an equation that is always true regardless of the values assigned to the variables. We use polynomial identities to expand or factorize polynomials.

We must learn polynomial identities in mathematics. Four important identities of the polynomial are listed below.

\(\color{blue}{\left(a\:+\:b\right)^2=\:a^2+\:2ab\:+\:b^2}\)
\(\color{blue}{\left(a\:−\:b\right)^2=\:a^2−\:2ab\:+\:b^2}\)
\(\color{blue}{\left(a\:+\:b\right)\left(a\:−\:b\right)\:=\:a^2−\:b^2}\)
\(\color{blue}{\left(x\:+\:a\right)\left(x\:+\:b\right)\:=\:x^2+\:x\left(a\:+\:b\right)\:+\:ab}\)

Apart from the simple polynomial identities mentioned above, there are other identities of polynomials. Here are some of the most common polynomial identities used:

\(\color{blue}{\left(a\:+\:b\:+\:c\right)^2=\:a^2+\:b^2+\:c^2+\:2ab\:+\:2bc\:+\:2ca}\)
\(\color{blue}{\left(a\:+\:b\right)^3=\:a^3+\:3a^2b\:+\:3ab^2+\:b^3}\)
\(\color{blue}{\left(a\:−\:b\right)^3=\:a^3−\:3a^2b+\:3ab^2−\:b^3}\)
\(\color{blue}{\left(a\right)^3+\:\left(b\right)^3=\:\left(a\:+\:b\right)\left(a^2−\:ab\:+\:b^2\right)}\)
\(\color{blue}{\left(a\right)^3−\:\left(b\right)^3=\:\left(a\:−\:b\right)\left(a^2+\:ab\:+\:b^2\right)}\)
\(\color{blue}{\left(a\right)^3+\:\left(b\right)^3+\:\left(c\right)^3−\:3abc\:=\:\left(a\:+\:b\:+\:c\right)\left(a^2+\:b^2+\:c^2−\:ab\:−\:bc−ca\right)}\)