Same Math › Algebra 1 · grades 8–10 › Factoring by grouping
What is factoring by grouping?
Factoring by grouping splits the middle term into two pieces, then factors the four terms in pairs. For 2x² + 7x + 3, a × c is 6, and 1 and 6 add to 7, so write 2x² + x + 6x + 3. The first pair gives x(2x + 1), the second gives 3(2x + 1), and the shared (2x + 1) pulls out: (2x + 1)(x + 3).
- Split the middle using two numbers that multiply to a × c.
- Pull out the GCF of each pair.
- Both pairs share a matching binomial. That's one factor.
Why schools teach it
It shows why every factoring shortcut works
Grouping is the full, written-out version of factoring ax² + bx + c (HSA.SSE.A.2). It uses only the GCF and the distributive property, which kids already know, so each step can be justified.
It also handles four-term expressions like x³ + 2x² + 3x + 6 that come up later, and it's the method many textbooks and tests show in their answer keys.
At the kitchen table
What to say, and what to skip
Say this
What's a times c?
Which two numbers multiply to that and add to b?
What does each pair have in common?
Pull it out.Do the two parentheses match?
If not, check the signs.
Skip this
- Pulling out a positive GCF when the third term is negative.For −6x − 3, pull out −3 to get −3(2x + 1).
- Stopping at x(2x + 1) + 3(2x + 1).That's a sum, not factored yet. Pull out (2x + 1).
- Skipping the GCF of the whole thing first.
See it in action
Factoring 3x² + 10x + 8
a × c = 24. Find a pair that adds to 10
4 and 6Split the middle
3x² + 6x + 4x + 8Pull out each pair's GCF
3x(x + 2) + 4(x + 2)Pull out the shared binomial
(x + 2)(3x + 4)
The order of the split matters only for the look. Either order gives the same answer.
Free practice sheet
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Parents also ask
Does it matter which number goes first when I split?
No. 2x² + 6x + x + 3 groups to 2x(x + 3) + 1(x + 3), the same answer.
What if the parentheses don't match?
Usually a sign slip: when the third term is negative, pull out a negative GCF from the second pair.
What is the AC method?
Another name for this: use a × c to find the two numbers, then group.
What about four terms with no x²?
Group them the same way. x³ + 2x² + 3x + 6 = x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3).