Factoring Trinomials Worksheets - Math Monks - Free Printable

We are given 8 trinomials (and one polynomial with 3 terms that is not quadratic, #3) to factor completely. Since the title says “Factoring Trinomials (a > 1)”, we expect most of these to be quadratics where the leading coefficient is greater than 1.

We’ll solve each one step by step using factoring techniques: GCF first, then AC method (or trial and error) for quadratics.

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Problem 1: 6x² + 37x + 6

- a = 6, b = 37, c = 6
- Multiply a*c = 6*6 = 36
- Find two numbers that multiply to 36 and add to 37 → 1 and 36
- Split middle term: 6x² + x + 36x + 6
- Group: (6x² + x) + (36x + 6)
- Factor each group:
- x(6x + 1) + 6(6x + 1)
- Factor out common binomial: (x + 6)(6x + 1)

✔ Answer: (x + 6)(6x + 1)

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Problem 2: 3b² - 3b - 36

- First, factor out GCF: 3(b² - b - 12)
- Now factor b² - b - 12
- Need two numbers multiplying to -12 and adding to -1 → -4 and 3
- So: (b - 4)(b + 3)
- Put back the 3: 3(b - 4)(b + 3)

✔ Answer: 3(b - 4)(b + 3)

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Problem 3: 4x³ + 43x² + 30x

- This is NOT a trinomial in the usual sense — it’s a cubic! But still, factor completely.
- First, factor out GCF: x(4x² + 43x + 30)
- Now factor 4x² + 43x + 30
- a=4, c=30 → ac=120
- Need two numbers multiplying to 120 and adding to 43 → 40 and 3
- Split: 4x² + 40x + 3x + 30
- Group: (4x² + 40x) + (3x + 30) = 4x(x + 10) + 3(x + 10)
- Factor: (4x + 3)(x + 10)
- So full factorization: x(4x + 3)(x + 10)

✔ Answer: x(4x + 3)(x + 10)

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Problem 4: 2p² + 14p + 24

- GCF: 2(p² + 7p + 12)
- Factor p² + 7p + 12 → two numbers multiplying to 12, adding to 7 → 3 and 4
- So: (p + 3)(p + 4)
- Full: 2(p + 3)(p + 4)

✔ Answer: 2(p + 3)(p + 4)

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Problem 5: -6y² - 25y - 25

- First, factor out -1 to make leading coefficient positive: -(6y² + 25y + 25)
- Now factor 6y² + 25y + 25
- ac = 6*25 = 150
- Need two numbers multiplying to 150, adding to 25 → 15 and 10
- Split: 6y² + 15y + 10y + 25
- Group: (6y² + 15y) + (10y + 25) = 3y(2y + 5) + 5(2y + 5)
- Factor: (3y + 5)(2y + 5)
- Don’t forget the negative sign: -(3y + 5)(2y + 5)

✔ Answer: -(3y + 5)(2y + 5)
*(Alternatively, you can write as (-3y -5)(2y + 5) or (3y+5)(-2y-5), but the above is cleanest.)*

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Problem 6: 12x² + 15x + 3

- GCF: 3(4x² + 5x + 1)
- Factor 4x² + 5x + 1
- ac = 4*1 = 4
- Numbers multiplying to 4, adding to 5 → 4 and 1
- Split: 4x² + 4x + x + 1
- Group: 4x(x + 1) + 1(x + 1) = (4x + 1)(x + 1)
- Full: 3(4x + 1)(x + 1)

✔ Answer: 3(4x + 1)(x + 1)

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Problem 7: 12b² - 24b + 9

- GCF: 3(4b² - 8b + 3)
- Factor 4b² - 8b + 3
- ac = 4*3 = 12
- Numbers multiplying to 12, adding to -8 → -6 and -2
- Split: 4b² - 6b - 2b + 3
- Group: (4b² - 6b) + (-2b + 3) = 2b(2b - 3) -1(2b - 3)
- Factor: (2b - 1)(2b - 3)
- Full: 3(2b - 1)(2b - 3)

✔ Answer: 3(2b - 1)(2b - 3)

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Problem 8: 4x² - 35x + 49

- a=4, c=49 → ac=196
- Need two numbers multiplying to 196 and adding to -35 → -28 and -7
- Split: 4x² - 28x - 7x + 49
- Group: (4x² - 28x) + (-7x + 49) = 4x(x - 7) -7(x - 7)
- Factor: (4x - 7)(x - 7)

✔ Answer: (4x - 7)(x - 7)

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## ✔ Final Answers:

1. (x + 6)(6x + 1)
2. 3(b - 4)(b + 3)
3. x(4x + 3)(x + 10)
4. 2(p + 3)(p + 4)
5. -(3y + 5)(2y + 5)
6. 3(4x + 1)(x + 1)
7. 3(2b - 1)(2b - 3)
8. (4x - 7)(x - 7)

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Let me know if you’d like to see any step explained in more detail!