Factoring Trinomials with a Leading Coefficient
I can factor a trinomial whose leading coefficient is not 1 by splitting the middle term.
The problem this sheet solves
The rule from the previous sheet was TRUE there: when the leading coefficient is 1, the pair comes from c. When it is not, looking at c alone finds nothing and the student concludes the trinomial does not factor. Nothing in their work is visibly wrong, which is why this one survives a whole unit. The fix is one multiplication and the area box says why. In the 2x2 box the two middle cells add to b and multiply to a times c, so the two jobs are unchanged — only the product target moved, from c to a times c. The sheet says out loud that the previous sheet was the special case where those two numbers coincide, so a student keeps one idea instead of two.
The key rule
Find the pair that multiplies to a·c and adds to b, split the middle term into those two, then read the factors off the edges of the box.
a·c on top. b in the middle.
Watch out
The mistake: For 3x2 + 7x + 2 a student says the pairs of 2 add to 3, never 7, so it does not factor.
The fix: It factors. The target is a·c = 3 · 2 = 6, not 2, and 1 and 6 add to 7. Splitting gives 3x2 + x + 6x + 2 = (3x + 1)(x + 2).
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.