Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each trinomial step by step. We’re factoring trinomials where the coefficient of $x^2$ (called “a”) is greater than 1. That means we can’t just guess and check easily — we’ll use the AC method or splitting the middle term.

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Problem 1: $4x^2 + 4x - 35$

Step 1: Multiply a and c → $4 \times (-35) = -140$

Step 2: Find two numbers that multiply to -140 and add to +4 (the middle term).

Try: 14 and -10 → 14 × (-10) = -140, 14 + (-10) = 4 ✔

Step 3: Split the middle term:

$4x^2 + 14x - 10x - 35$

Step 4: Group:

$(4x^2 + 14x) + (-10x - 35)$

Factor each group:

$2x(2x + 7) -5(2x + 7)$

Now factor out the common binomial:

$(2x - 5)(2x + 7)$

✔ Check: $(2x - 5)(2x + 7) = 4x^2 + 14x - 10x - 35 = 4x^2 + 4x - 35$ ✔️

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Problem 2: $5x^2 + 37x - 24$

a×c = 5 × (-24) = -120

Find two numbers that multiply to -120 and add to +37.

Try: 40 and -3 → 40 × (-3) = -120, 40 + (-3) = 37 ✔

Split middle term:

$5x^2 + 40x - 3x - 24$

Group:

$(5x^2 + 40x) + (-3x - 24)$

Factor:

$5x(x + 8) -3(x + 8)$

→ $(5x - 3)(x + 8)$

✔ Check: $5x·x = 5x^2$, $5x·8=40x$, $-3·x=-3x$, $-3·8=-24$ → total: $5x^2 + 37x -24$ ✔️

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Problem 3: $3x^2 + 12x + 12$

First, factor out GCF if possible → all terms divisible by 3!

→ $3(x^2 + 4x + 4)$

Now factor inside: $x^2 + 4x + 4 = (x+2)^2$

So final answer: $3(x + 2)^2$

✔ Check: $3(x+2)(x+2) = 3(x^2 + 4x + 4) = 3x^2 + 12x + 12$ ✔️

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Problem 4: $6x^2 - 44x + 14$

GCF? All even → factor out 2 first:

→ $2(3x^2 - 22x + 7)$

Now factor $3x^2 - 22x + 7$

a×c = 3×7 = 21

Need two numbers that multiply to 21 and add to -22 → -21 and -1

Split: $3x^2 - 21x - x + 7$

Group: $(3x^2 - 21x) + (-x + 7)$

Factor: $3x(x - 7) -1(x - 7)$ → $(3x - 1)(x - 7)$

Don’t forget the 2 we pulled out!

Final: $2(3x - 1)(x - 7)$

✔ Check: $2[(3x)(x) + (3x)(-7) + (-1)(x) + (-1)(-7)] = 2[3x^2 -21x -x +7] = 2[3x^2 -22x +7] = 6x^2 -44x +14$ ✔️

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Problem 5: $3x^2 - 23x - 36$

a×c = 3×(-36) = -108

Need two numbers that multiply to -108 and add to -23.

Try: -27 and +4 → -27×4 = -108, -27+4 = -23 ✔

Split: $3x^2 -27x +4x -36$

Group: $(3x^2 -27x) + (4x -36)$

Factor: $3x(x - 9) +4(x - 9)$ → $(3x + 4)(x - 9)$

✔ Check: $3x·x=3x^2$, $3x·(-9)=-27x$, $4·x=4x$, $4·(-9)=-36$ → total: $3x^2 -23x -36$ ✔️

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Problem 6: $2x^2 - 3x + 1$

a×c = 2×1 = 2

Need two numbers that multiply to 2 and add to -3 → -1 and -2

Split: $2x^2 -2x -x +1$

Group: $(2x^2 -2x) + (-x +1)$

Factor: $2x(x - 1) -1(x - 1)$ → $(2x - 1)(x - 1)$

✔ Check: $2x·x=2x^2$, $2x·(-1)=-2x$, $-1·x=-x$, $-1·(-1)=1$ → total: $2x^2 -3x +1$ ✔️

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Problem 7: $20x^2 + 2x - 6$

GCF? All even → factor out 2:

→ $2(10x^2 + x - 3)$

Now factor $10x^2 + x - 3$

a×c = 10×(-3) = -30

Need two numbers that multiply to -30 and add to +1 → +6 and -5

Split: $10x^2 +6x -5x -3$

Group: $(10x^2 +6x) + (-5x -3)$

Factor: $2x(5x + 3) -1(5x + 3)$ → $(2x - 1)(5x + 3)$

Don’t forget the 2!

Final: $2(2x - 1)(5x + 3)$

✔ Check: $2[(2x)(5x) + (2x)(3) + (-1)(5x) + (-1)(3)] = 2[10x^2 +6x -5x -3] = 2[10x^2 +x -3] = 20x^2 +2x -6$ ✔️

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Problem 8: $3x^2 + 25x + 42$

a×c = 3×42 = 126

Need two numbers that multiply to 126 and add to 25 → 18 and 7

Split: $3x^2 +18x +7x +42$

Group: $(3x^2 +18x) + (7x +42)$

Factor: $3x(x + 6) +7(x + 6)$ → $(3x + 7)(x + 6)$

✔ Check: $3x·x=3x^2$, $3x·6=18x$, $7·x=7x$, $7·6=42$ → total: $3x^2 +25x +42$ ✔️

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Problem 9: $2x^2 - 3x - 20$

a×c = 2×(-20) = -40

Need two numbers that multiply to -40 and add to -3 → -8 and +5

Split: $2x^2 -8x +5x -20$

Group: $(2x^2 -8x) + (5x -20)$

Factor: $2x(x - 4) +5(x - 4)$ → $(2x + 5)(x - 4)$

✔ Check: $2x·x=2x^2$, $2x·(-4)=-8x$, $5·x=5x$, $5·(-4)=-20$ → total: $2x^2 -3x -20$ ✔️

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Problem 10: $35x^2 - 11x - 6$

a×c = 35×(-6) = -210

Need two numbers that multiply to -210 and add to -11 → -21 and +10

Split: $35x^2 -21x +10x -6$

Group: $(35x^2 -21x) + (10x -6)$

Factor: $7x(5x - 3) +2(5x - 3)$ → $(7x + 2)(5x - 3)$

✔ Check: $7x·5x=35x^2$, $7x·(-3)=-21x$, $2·5x=10x$, $2·(-3)=-6$ → total: $35x^2 -11x -6$ ✔️

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Final Answer:
1. $(2x - 5)(2x + 7)$
2. $(5x - 3)(x + 8)$
3. $3(x + 2)^2$
4. $2(3x - 1)(x - 7)$
5. $(3x + 4)(x - 9)$
6. $(2x - 1)(x - 1)$
7. $2(2x - 1)(5x + 3)$
8. $(3x + 7)(x + 6)$
9. $(2x + 5)(x - 4)$
10. $(7x + 2)(5x - 3)$