Factoring

Greatest Common Factor

Factor a polynomial by pulling out the greatest common factor (GCF) from all terms. Find the largest monomial that divides every term, factor it out, and verify by expanding.

gcf The GCF of a polynomial is the largest factor shared by every term. After factoring out the GCF, expand to verify the factored form equals the original.

Example

Factor out the greatest common factor shared by every term.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

6x2+9x6x^{2} + 9x

Step 2 — Find GCF

Find the greatest common factor: 3x.

GCF=3x\text{GCF}= \hlmath{3x}

Step 3 — Divide terms

Divide coefficients: 6 / 3 = 2 and 9 / 3 = 3.

6/3=29/3=36 / 3 = \hl{2} \quad 9 / 3 = \hl{3}

Step 4 — Factored form

Write the factored form: 3x times (2x + 3).

3x(2x+3)\hlmath{3x} ( \hlmath{2x} + \hl{3} )

Step 5 — Check

Check by expanding back to the original.

3x(2x+3)=6x2+9x3x ( 2x + 3 )= 6x^{2} + 9x