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All courses Math III · A-APR.4 4 of 55
Prove polynomial identities and use them to solve or describe numerical relationships.

Verify a polynomial identity by expanding and comparing both sides

Problem
Verify this polynomial identity by expanding one or both sides: (x + 2)² = x² + 4x + 4.
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Hint

Expand the left side \((x+2)^2\) using the binomial square pattern or direct multiplication.

\((x+2)^2 = (x+2)(x+2)\), so multiply each term and combine like terms.

Solution walkthrough

01

Write the square as a product

\[(x+2)^2~=~(x+2)(x+2)\]

To verify the identity, expand the left side into ordinary multiplication.

02

Multiply the binomials

\[(x+2)(x+2)~=~x^2~+~2x~+~2x~+~4\]

Distribute each term in the first binomial across the second binomial.

03

Combine like terms

\[x^2~+~2x~+~2x~+~4~=~x^2~+~4x~+~4\]

The two middle terms add to \(4x\).

04

Compare both sides

\[x^2~+~4x~+~4~=~x^2~+~4x~+~4\]

Since the expansion of the left side matches the right side exactly, the identity is verified.

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Another way

  1. You can use the pattern \((a+b)^2 = a^2 + 2ab + b^2\) with \(a = x\) and \(b = 2\).

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Common mistake

A common mistake is to write \((x+2)^2 = x^2 + 4\). The middle term \(2(x)(2) = 4x\) must be included.