Factoring polynomialsEasy
Name: Date:
Factor each one. The number in front of x² is not 1.
13x² − 3
22x² + 10x + 8
32x² + x − 10
42x² − x − 1
52x² − 8
62x² − 4x − 6
73x² + x − 10
82x² − 4x + 2
Factoring trinomials with a leading coefficient worksheets
Once the number in front of x² is not 1, “two numbers that multiply to c and add to b” stops working on its own. These free printable worksheets factor trinomials like 2x² + x − 10 and 6x² + 17x − 10, and the answer key splits the middle term and groups, line by line.
If they get stuck
Factoring trinomials with a leading coefficient
The answer key prints on a separate page.
What's on this sheet
The first question on every one of these is not about brackets at all: is there a number that divides all three terms? 3x² − 3 is 3(x² − 1) before it is anything else, and 2x² + 10x + 8 is 2(x² + 5x + 4). About a third of each sheet is built that way on purpose, because a student who skips the common factor is left searching for two brackets that multiply to 2x² + 10x + 8 — which exist, and are much harder to find.
Why this sheet works
The rest have a leading coefficient that stays in a bracket: (2x + 1)(x − 1), (3x + 1)(x − 3). The key works these by splitting the middle term. For 2x² + x − 10, a × c is −20; the pair −4 and 5 multiplies to −20 and adds to 1; so x becomes −4x + 5x, and grouping the four terms gives 2x(x − 2) + 5(x − 2) = (x − 2)(2x + 5). Every line of that is on the key, because guessing brackets gets slow once a is 4 or 6, and this method does not.
How this one works
3x² − 3
3(x² − 1)
x² − 1:a × c = 1 × (−1) = −1
1 × (−1) = −11 + (−1) = 0
x² + x − x − 1
x(x + 1) − 1(x + 1)
(x + 1)(x − 1)
3x² − 3 = 3(x − 1)(x + 1)
- Every coefficient shares 3. Take it out in front — it stays there to the end and the rest is a simpler polynomial.
- Take the quadratic x² − 1. Multiply the number in front of x² by the last number: 1 × −1 = −1.
- Find two numbers that multiply to −1 and add to 0, the middle coefficient. They are 1 and −1.
- Split the middle term using them: 0x becomes 1x and −1x.
- Take the common factor out of each pair. The same bracket, (x + 1), comes out both times.
- Both parts share (x + 1), so take it out front. What is left, x − 1, is the other bracket.
- Collect the brackets: 3x² − 3 = 3(x − 1)(x + 1). Multiply back out to check.