Unit 6 · Lesson 1 of 6

Factoring out the GCF and factoring by grouping

About 25 minutes · 5 steps · one idea: pull out what every term shares

You'll be able to: factor out the greatest common factor of a polynomial, and factor a four-term polynomial by grouping.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.

Multiply out: \(-4x(2x - 3)\).

Multiply \(-4x\) by each term: \(-4x \cdot 2x = -8x^2\) and \(-4x \cdot (-3) = 12x\).

Answer: \(-8x^2 + 12x\).

Simplify the fraction \(\dfrac{18}{24}\).

The greatest number that divides both 18 and 24 is 6.

\(\dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}\).

Step 2 of 5The idea

To factor means to rewrite a sum as a product. It is multiplying in reverse: \(-4x(2x - 3)\) becomes \(-8x^2 + 12x\), and factoring takes you back.

The GCF (greatest common factor) of the terms is the biggest thing that divides every term. For the number part, use the biggest number that divides every coefficient. For each variable, use the lowest power that appears in every term.

Factor out the GCF. Find the GCF, write it in front, and divide each term by it to fill the parentheses: \(6x^3 + 9x^2 = 3x^2(2x + 3)\).

Negative GCF. If the first term is negative, factor out the negative. Every sign inside flips: \(-5x^2 + 10x = -5x(x - 2)\).

A binomial can be the GCF. \(x(x + 4) + 3(x + 4) = (x + 4)(x + 3)\).

Grouping (4 terms). Group the first two and the last two terms. Factor the GCF out of each pair. If the two parentheses match, factor that binomial out.

You can always check by multiplying back out. You should get the original polynomial.

Why this shows up on the placement test: pulling out the GCF is the first step of every factoring problem, and grouping is how harder trinomials get factored in the next lessons.

Step 3 of 5Worked examples

Example 1: \(12x^3 - 18x^2 + 6x\)

  1. Numbers 12, 18, 6: the biggest number dividing all three is 6.
  2. Variables \(x^3, x^2, x\): the lowest power is \(x\). So the GCF is \(6x\).
  3. Divide each term: \(\dfrac{12x^3}{6x} = 2x^2\), \(\dfrac{-18x^2}{6x} = -3x\), \(\dfrac{6x}{6x} = 1\).
  4. Answer: \(6x(2x^2 - 3x + 1)\). Check: \(6x \cdot 2x^2 - 6x \cdot 3x + 6x \cdot 1 = 12x^3 - 18x^2 + 6x\).

Example 2: \(2x^3 - 6x^2 - 5x + 15\) by grouping

  1. Group in pairs: \((2x^3 - 6x^2) + (-5x + 15)\).
  2. First pair: GCF \(2x^2\), so \(2x^2(x - 3)\).
  3. Second pair: the first term is negative, so take out \(-5\): \(-5(x - 3)\). Taking out \(-5\) makes the parentheses match the first pair.
  4. Now \(2x^2(x - 3) - 5(x - 3)\). The common factor is \((x - 3)\).
  5. Answer: \((x - 3)(2x^2 - 5)\). Check: \(2x^3 - 5x - 6x^2 + 15\), the same four terms.

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. Type products with parentheses, like 3x(x - 2) or (x + 1)(x^2 + 4).

Factor out the GCF: \(15x^2 + 10x\).

Numbers 15 and 10: GCF 5. Variables \(x^2\) and \(x\): lowest power \(x\). GCF \(= 5x\).

\(\dfrac{15x^2}{5x} = 3x\) and \(\dfrac{10x}{5x} = 2\).

Answer: \(5x(3x + 2)\).

Factor out the GCF: \(8x^4 - 12x^3 + 20x^2\).

Numbers 8, 12, 20: GCF 4. Lowest power of \(x\): \(x^2\). GCF \(= 4x^2\).

Divide each term: \(2x^2\), \(-3x\), \(5\).

Answer: \(4x^2(2x^2 - 3x + 5)\).

Factor out the GCF: \(6x^3y^2 - 9x^2y^3\).

Numbers 6 and 9: GCF 3. Lowest powers: \(x^2\) and \(y^2\). GCF \(= 3x^2y^2\).

\(\dfrac{6x^3y^2}{3x^2y^2} = 2x\) and \(\dfrac{-9x^2y^3}{3x^2y^2} = -3y\).

Answer: \(3x^2y^2(2x - 3y)\).

Factor out a negative GCF: \(-4x^3 + 8x^2 - 12x\).

Numbers 4, 8, 12: GCF 4. Lowest power: \(x\). The first term is negative, so use \(-4x\).

Divide each term by \(-4x\): \(x^2\), \(-2x\), \(+3\). Dividing by a negative flips each sign.

Answer: \(-4x(x^2 - 2x + 3)\).

Factor: \(7x(x - 5) + 2(x - 5)\).

Both terms share the binomial \((x - 5)\). Take it out.

What is left from each term: \(7x\) and \(+2\).

Answer: \((x - 5)(7x + 2)\).

Factor by grouping: \(x^3 - 2x^2 + 5x - 10\).

Group: \((x^3 - 2x^2) + (5x - 10)\).

Each pair: \(x^2(x - 2) + 5(x - 2)\).

Answer: \((x - 2)(x^2 + 5)\).

Factor by grouping: \(3x^3 + 6x^2 - 4x - 8\).

Group: \((3x^3 + 6x^2) + (-4x - 8)\).

First pair: \(3x^2(x + 2)\). Second pair: take out \(-4\) to get \(-4(x + 2)\).

Answer: \((x + 2)(3x^2 - 4)\).

Which is the factored form of \(2xy - 6x + 5y - 15\)?

  1. \((y + 3)(2x - 5)\)
  2. \((y - 3)(2x + 5)\)
  3. \((y - 3)(2x - 5)\)
  4. \((y + 3)(2x + 5)\)

Group: \((2xy - 6x) + (5y - 15)\).

First pair: \(2x(y - 3)\). Second pair: \(5(y - 3)\).

Answer: \((y - 3)(2x + 5)\), choice (b). Check: \(2xy + 5y - 6x - 15\), the same four terms.

Multiplying out the others gives a wrong sign on at least one term, for example (c) gives \(2xy - 5y - 6x + 15\).

Step 5 of 5Wrap-up

You can now factor out a GCF (including a negative one or a whole binomial) and factor four terms by grouping them in pairs.

Read or watch next: OpenStax Intermediate Algebra 2e, 6.1 Greatest Common Factor and Factor by Grouping (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Algebra 1: Intro to factoring (short videos). Quick rules: Factoring cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "In grouping, what do I do if the two parentheses don't match?" It knows this course and can explain things another way.