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Factoring 3x2+6bx−3ax−6ab3x^2 + 6bx - 3ax - 6ab

Factor 3x2+6bx−3ax−6ab3x^2 + 6bx - 3ax - 6ab completely by extracting the GCF and then applying factoring by grouping to the remaining four-term polynomial.

Step 1 — Is there a GCF? Yes, the GCF of 3x23x^2, 6bx6bx, 3ax3ax, and 6ab6ab is 33. Factor it out:

3x2+6bx−3ax−6ab=3(x2+2bx−ax−2ab)3x^2 + 6bx - 3ax - 6ab = 3(x^2 + 2bx - ax - 2ab)

Step 2 — Classify the expression inside the parentheses. The expression x2+2bx−ax−2abx^2 + 2bx - ax - 2ab has four terms, so use factoring by grouping.

Step 3 — Group and factor each pair:

3[x(x+2b)−a(x+2b)]3[x(x + 2b) - a(x + 2b)]

Both groups share the common binomial factor (x+2b)(x + 2b). Factor it out:

3(x+2b)(x−a)3(x + 2b)(x - a)

Step 4 — Check.

  • Is the expression factored completely? Yes — neither binomial can be factored further.
  • Verify by multiplying: 3(x+2b)(x−a)=3(x2−ax+2bx−2ab)=3x2−3ax+6bx−6ab3(x + 2b)(x - a) = 3(x^2 - ax + 2bx - 2ab) = 3x^2 - 3ax + 6bx - 6ab ✓.

The completely factored form is 3(x+2b)(x−a)3(x + 2b)(x - a). This example combines two factoring techniques within the general strategy: first extracting a numerical GCF from all four terms, and then applying the grouping method to the resulting four-term polynomial inside the parentheses. Without factoring out the GCF of 33 first, the coefficients inside the polynomial would be larger, making the grouping step more difficult to recognize.

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Updated 2026-04-21

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References

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OpenStax

Elementary Algebra @ OpenStax

Ch.7 Factoring - Elementary Algebra @ OpenStax

Algebra

Math

Prealgebra

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