Factoring Out the Greatest Common Factor
I can factor the greatest common factor out of a polynomial and check it by multiplying back.
The problem this sheet solves
When a term IS the GCF, what stays behind in its place is 1, not nothing. The student has found the correct GCF and divided the first term correctly, so there is nothing on the page to mark. The other half of the unit's trouble starts here too: a common factor is not the greatest one, and 6x out of 24x cubed plus 36x squared is TRUE and unfinished. Factoring is distribution run backwards, so the check is already in the student's hands: multiply back out. 7x times x gives 7x squared, which is missing a whole term, and a student can see that without being told a rule. Every later sheet in this unit checks its answer the same way, so the habit is worth more here than the procedure. The area box is used in reverse — the product in the cells, the GCF on one edge, and the other edge is what to find.
The key rule
Take out the largest number and the highest power that divide EVERY term, then multiply back to check. A term that IS the GCF leaves a 1 behind, not a blank.
Multiply back. It must match.
Watch out
The mistake: A student writes 7x2 + 7x = 7x(x).
The fix: 7x(x + 1). The second term IS the GCF, so a 1 stays behind in its place. Multiply back: 7x(x) gives only 7x2, and the 7x has vanished. At x = 1 the answer is 14, and theirs is 7.
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.