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All courses Algebra II · A-SSE.2 23 of 55
Use expression structure to find useful rewrites.

Factor a polynomial by grouping to reveal a common binomial or polynomial factor

Problem
Factor \(x^3~+~2x^2~+~3x~+~6\) completely by grouping.
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Hint

Group the first two terms and the last two terms, then look for a common binomial factor.

Factoring by grouping works when each pair has a common factor and both groups simplify to the same remaining expression.

Solution walkthrough

01

Group adjacent terms

\[(x^3+2x^2)+(3x+6)\]

The first pair shares x squared, and the second pair shares 3.

02

Factor each group

\[x^2(x+2)+3(x+2)\]

Factoring the greatest common factor from each pair reveals the same binomial x plus 2.

03

Factor the common binomial

\[(x+2)(x^2+3)\]

Treat x plus 2 as the shared factor; the remaining terms are x squared and 3.

04

Verify complete factorization

\[(x+2)(x^2+3)=x^3+2x^2+3x+6~->~(x+2)(x^2+3)\]

Expansion returns the original polynomial, and x squared plus 3 has no real or integer factorization, so the factorization is complete, choice A.

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Another way

  1. Check x equals negative 2 as a root, divide by x plus 2, and obtain the quotient x squared plus 3.

!

Common mistake

Both groups must expose the identical binomial; factoring 3x plus 6 as 3(x plus 2) is what makes grouping work.

Solution walkthrough video