LEMMA

Polynomials & quadratics

Factor the greatest common factor

ALEKS placement

Find what both terms contain, then write that factor once outside parentheses.

What this covers

  • Find the greatest common factor of the coefficients.
  • Take the lowest exponent of each variable that appears in every term.
  • Divide each term by the GCF and write the quotients in parentheses.
  • Multiply back mentally to check.

Worked example

Worked example

Factor 6x³ − 9x²

  1. 6=3⋅2Start with the numbers. Three divides six evenly.
  2. −9=3⋅(−3)Three also divides negative nine. It is the greatest positive number shared by both coefficients.
  3. x3=x2⋅xThe first term contains three copies of the variable. Both terms contain at least two copies.
  4. 6x3=3x2(2x)Take out the shared number and shared variable factors. The first term leaves two times the variable.
  5. −9x2=3x2(−3)The second term leaves negative three. Its negative sign stays attached.
  6. 6x3−9x2=3x2(2x−3)Write the shared factor once outside the group. Multiplying it into each term restores the original expression, including at a zero input.
Another worked example

Factor 5x(x − 2) + 3(x − 2).

  1. 5x(x−2)+3(x−2)=(x−2)(5x+3)Both terms contain the entire group x − 2. Factor that group out as one unit, leaving 5x and 3 behind.
  2. (x−2)(5x+3)=5x2−7x−6Expanding gives 5x² + 3x − 10x − 6, which is also what the original expands to.

A common mistake

A mistake Lemma catches

From 6x3−9x2

6x3−9x2=3x2(2x2−3)6x3−9x2=3x2(2x−3)

“Each term inside is the original term divided by the GCF 3x²: 6x³ ÷ 3x² = 2x (the powers subtract, 3 − 2 = 1) and −9x² ÷ 3x² = −3. Multiply back to check: 3x²(2x² − 3) = 6x⁴ − 9x², not 6x³ − 9x².”

Try one

Sample problem

Factor completely.

4x+10

Practice factor gcf free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this