Factoring by Grouping
I can factor a four term polynomial by grouping, and use the matching brackets as a check.
The problem this sheet solves
Grouping is the only method in the unit that tells you MID-CALCULATION whether you are right, and students do not know to look. The two brackets must be identical; if they are not, the wrong factor came out — nearly always the sign — and there is no point going on. That diagnostic is a stronger habit than multiplying back, because it stops the error where it happens rather than at the end. Every polynomial here has four terms to begin with and none came from a trinomial, which is the case grouping exists for. Several answers are correct and unfinished — one bracket is still a difference of squares — while two others hide a sum of squares that does not factor, so students have to decide rather than pattern-match.
The key rule
Take the GCF out of each pair so both leave the SAME bracket. If the two brackets differ, the sign of the second GCF is wrong.
Same bracket, or start again.
Watch out
The mistake: For 2x³ + 6x² − 5x − 15 a student writes 2x²(x + 3) − 5(x − 3).
The fix: 2x²(x + 3) − 5(x + 3). Taking out −5 rather than 5 keeps the bracket the same. Their brackets differ, which is the signal to stop — and −5(x − 3) is −5x + 15, not −5x − 15.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.