MCF3M Functions & Applications · Unit 2 — The Algebra of Quadratic Expressions
Lesson 2.2
Factoring Using the Greatest Common Factor
Run expanding in reverse: find the greatest common factor, move it outside a bracket, and expand to check that the factorization is correct and complete.
Estimated time: 30–45 minutes
Learning goals
- explain factoring as the reverse of expanding
- find the GCF of numbers and variables
- factor expressions completely, including negative and binomial common factors
- verify a factorization by expanding
Step 1 of 4
Warm Up
Success criteria
Real-world idea
Packing snack bags
You have 18 granola bars and 24 juice boxes, and every bag must be identical. The most bags you can make is 6 — the greatest common factor.
Factoring algebra works the same way: find the biggest identical piece every term shares.
Step 2 of 4
Learn
Factoring undoes expanding
Expanding: 5x(x − 4) → 5x² − 20x. Factoring goes back: 5x² − 20x → 5x(x − 4).
Why does it work? The distributive property read backwards: ab + ac = a(b + c). If every term has a factor a, we can write a once outside the bracket.
Key term
Greatest common factor (GCF)
The largest number and highest power of each variable that divides every term.
GCF of 12x² and 18x is 6x.
Key term
Factored completely
No factor other than 1 can still be taken out of any bracket.
Algebra workspace · Factor Builder
Choose a number part and a variable part, move them outside the bracket, build what's left, then expand to verify.
Predict first
For 15x² − 25x, a classmate pulls out only 5. Will 5(3x² − 5x) be factored completely?
Make a prediction to open the workspace.
Worked example
Factor completely 12x² + 18x.
Numbers
The greatest number dividing 12 and 18 is 6.
GCF(12, 18) = 6
Variables
x² and x share one x (the lowest power).
x
Divide each term
12x² ÷ 6x = 2x and 18x ÷ 6x = 3.
6x(2x + 3)
Check
Expand: 6x · 2x + 6x · 3 = 12x² + 18x ✓. 2x + 3 share nothing, so it is complete.
Answer: 6x(2x + 3) — 6x is the GCF because it is the biggest number and highest power of x in both terms.
Negative common factors
Worked example
Factor 5x(x + 2) − 3(x + 2).
Spot the shared piece
Both parts are multiplied by the same bracket (x + 2). Treat the whole bracket like one block, say B.
5x·B − 3·B
Factor it out
B(5x − 3), then put the bracket back.
(x + 2)(5x − 3)
Check
(x + 2)(5x − 3) = 5x² + 7x − 6, and 5x(x + 2) − 3(x + 2) = 5x² + 10x − 3x − 6 ✓
Answer: (x + 2)(5x − 3). The common factor is the entire binomial, because x + 2 multiplies both 5x and −3.
Step 3 of 4
Practice
Build it
Factor out the GCF.
Apply it
Challenge
Explain it
Step 4 of 4
Mastery Check
Lesson 2.2 mastery check
6 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.