Factoring Out the GCF

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Factoring Out the GCF: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Factoring out the GCF means pulling the greatest common factor of all terms in front of parentheses, so 12x³ + 18x² − 30x becomes 6x(2x² + 3x − 5).

Algebra 1AlgebraA-SSE.A.2

What is Factoring Out the GCF?

When every term of an expression shares a number or a letter, factoring out the GCF moves that shared piece outside parentheses and leaves the remainder inside. It is the distributive property read right to left.

The shared piece combines the biggest divisor of the coefficients with the smallest power of every letter present in all terms. With it removed, the parentheses hold the leftovers, and multiplying back must restore the original sum.

How to spot it

  • Factoring out the GCF is possible when every term is divisible by one number, or contains one letter.
  • A direction says factor completely and the terms look like they share something.
  • A quadratic like \(3x^2 + 6x\) has no constant term and needs its x pulled out.
  • An equation is set to zero and must be solved by factoring first.

The method

  1. Find the biggest whole number that goes into every coefficient.
  2. For each letter present in all terms, keep its smallest exponent.
  3. Multiply those pieces to form the GCF, then divide every term by it when factoring out the GCF.
  4. Put the quotients inside parentheses, keeping signs, and multiply back to check.
12x³18x²−30x2x²3x−5divide each term by the GCF, 6x= 6x(2x² + 3x − 5)
Dividing each term by 6x leaves 2x², 3x and −5 inside the parentheses.

Factoring Out the GCF: solved example

Problem

Factor \(24a^2b – 36ab^2\) completely.

Solution

The largest number dividing 24 and 36 is 12. Both terms contain a and b, with smallest exponents 1 and 1, so the GCF is \(12ab\).

Divide: \(24a^2b \div 12ab = 2a\) and \(-36ab^2 \div 12ab = -3b\). The result is \(12ab(2a – 3b)\).

Try it

Question 1

Factor \(15x^2 + 10x\).

Show the answer

The GCF is \(5x\): \(5x(3x + 2)\).

Question 2

Factor \(8y^3 – 12y^2 + 4y\).

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The GCF is \(4y\): \(4y(2y^2 – 3y + 1)\).

Factoring Out the GCF vs. Greatest Common Factor

The greatest common factor is a number or expression, the thing you find. Taking it out of a sum is the factoring step that uses it. The first is the tool and the second is the action, and it is performed before any other factoring technique.

Greatest Common Factor (GCF)

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