Math worksheet for factoring polynomials using GCF and quadratic expressions, featuring 12 problems to solve.

Let’s go through each problem one by one. We’re factoring polynomials — that means we’re rewriting them as a product of simpler expressions. The first step is always to look for the Greatest Common Factor (GCF) — the biggest number and variable that divides all terms. After pulling out the GCF, if what’s left is a quadratic (like ax² + bx + c), we try to factor it further.

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Problem 1: 2ab² - 14ab

Step 1: Find GCF of coefficients: 2 and 14 → GCF = 2
Variables: both have at least one ‘a’ and one ‘b’ → so GCF includes ab
→ GCF = 2ab

Step 2: Divide each term by 2ab:
2ab² ÷ 2ab = b
-14ab ÷ 2ab = -7

So:
2ab(b - 7)

✔ Check: 2ab * b = 2ab², 2ab * (-7) = -14ab → correct.

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Problem 2: 5a² + 55a

GCF of 5 and 55 → 5
Both have ‘a’ → GCF = 5a

Divide:
5a² ÷ 5a = a
55a ÷ 5a = 11

So:
5a(a + 11)

✔ Check: 5a*a = 5a², 5a*11 = 55a → correct.

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Problem 3: 18v² + 114v - 84

First, find GCF of 18, 114, 84.

Factors:
- 18: 1,2,3,6,9,18
- 114: divisible by 2,3,6 → 114÷6=19
- 84: divisible by 6 → 84÷6=14

GCF = 6

Factor out 6:
6(3v² + 19v - 14)

Now factor the quadratic: 3v² + 19v - 14

We need two numbers that multiply to 3*(-14) = -42 and add to 19.

Try: 21 and -2 → 21 * (-2) = -42, 21 + (-2) = 19 ✔

Split middle term:
3v² + 21v - 2v - 14

Group:
(3v² + 21v) + (-2v -14)
= 3v(v + 7) -2(v + 7)
= (3v - 2)(v + 7)

So full answer:
6(3v - 2)(v + 7)

✔ Check: Expand back? Let’s skip for now but logic holds.

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Problem 4: 8x² + 16x

GCF of 8 and 16 → 8
Both have x → GCF = 8x

Divide:
8x² ÷ 8x = x
16x ÷ 8x = 2

So:
8x(x + 2)

✔ Simple check works.

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Problem 5: 25mp² - 45mp

Coefficients: 25 and 45 → GCF = 5
Variables: both have m and p → lowest power is p¹ → GCF = 5mp

Divide:
25mp² ÷ 5mp = 5p
-45mp ÷ 5mp = -9

So:
5mp(5p - 9)

✔ Check: 5mp*5p = 25mp², 5mp*(-9) = -45mp → good.

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Problem 6: 16c² + 48c

GCF of 16 and 48 → 16
Both have c → GCF = 16c

Divide:
16c² ÷ 16c = c
48c ÷ 16c = 3

So:
16c(c + 3)

✔ Correct.

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Problem 7: m²y + 6myn + 8n²y

All terms have ‘y’ → factor out y first:

y(m² + 6mn + 8n²)

Now factor the trinomial: m² + 6mn + 8n²

Look for two numbers that multiply to 8 and add to 6 → 2 and 4

So: (m + 2n)(m + 4n)

Full answer:
y(m + 2n)(m + 4n)

✔ Check: (m+2n)(m+4n) = m² + 4mn + 2mn + 8n² = m² + 6mn + 8n² → yes.

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Problem 8: -2x² + 8xy + 64y²

First, notice leading coefficient is negative. Factor out -2 to make it easier:

-2(x² - 4xy - 32y²)

Now factor inside: x² - 4xy - 32y²

Need two numbers that multiply to -32 and add to -4 → -8 and +4

So: (x - 8y)(x + 4y)

Full answer:
-2(x - 8y)(x + 4y)

✔ Check: (x-8y)(x+4y) = x² +4xy -8xy -32y² = x² -4xy -32y² → then times -2 gives original → correct.

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Problem 9: -42x² + 306xy - 84y²

First, factor out GCF. All coefficients divisible by 6? Let’s see:

42, 306, 84 → all divisible by 6? Yes. But also, since leading term is negative, let’s factor out -6 to make leading coefficient positive.

Check: -42 ÷ (-6) = 7
306 ÷ (-6) = -51
-84 ÷ (-6) = 14

Wait — better to factor out greatest common factor including sign.

Actually, GCF of 42, 306, 84:

Find GCF of 42, 306, 84.

42 = 2×3×7
306 = 2×3×3×17
84 = 2×2×3×7
Common: 2×3 = 6

But since first term is negative, factor out -6:

-6(7x² - 51xy + 14y²)

Now factor 7x² - 51xy + 14y²

Multiply 7 × 14 = 98. Need two numbers that multiply to 98 and add to -51 → -49 and -2

Because (-49)*(-2)=98, (-49)+(-2)= -51

Split middle term:

7x² - 49xy - 2xy + 14y²

Group:

(7x² - 49xy) + (-2xy + 14y²)
= 7x(x - 7y) -2y(x - 7y)
= (7x - 2y)(x - 7y)

So full answer:
-6(7x - 2y)(x - 7y)

✔ Check: Multiply back? Skip for brevity, but steps are solid.

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Problem 10: -m²v - 8mvn - 12n²v

All terms have ‘v’ → factor out v first:

v(-m² - 8mn - 12n²)

Now factor inside: -m² -8mn -12n² → factor out -1 to make leading coefficient positive:

= v[ -1(m² + 8mn + 12n²) ] = -v(m² + 8mn + 12n²)

Now factor m² + 8mn + 12n² → numbers that multiply to 12, add to 8 → 2 and 6

So: (m + 2n)(m + 6n)

Full answer:
-v(m + 2n)(m + 6n)

✔ Check: (m+2n)(m+6n) = m² + 6mn + 2mn + 12n² = m² + 8mn + 12n² → times -v → matches original.

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Problem 11: 2y² + 28y + 96

GCF of 2, 28, 96 → 2

Factor out 2:
2(y² + 14y + 48)

Factor trinomial: y² + 14y + 48 → numbers that multiply to 48, add to 14 → 6 and 8

So: (y + 6)(y + 8)

Full answer:
2(y + 6)(y + 8)

✔ Easy check.

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Problem 12: 5b²k² + 25bk² - 250k²

All terms have k² → factor out k² first.

Also, coefficients: 5, 25, 250 → GCF = 5

So overall GCF = 5k²

Factor out 5k²:

5k²(b² + 5b - 50)

Now factor b² + 5b - 50 → numbers that multiply to -50, add to 5 → 10 and -5

So: (b + 10)(b - 5)

Full answer:
5k²(b + 10)(b - 5)

✔ Check: (b+10)(b-5) = b² -5b +10b -50 = b² +5b -50 → times 5k² → matches.

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Final Answer:

1. 2ab(b - 7)
2. 5a(a + 11)
3. 6(3v - 2)(v + 7)
4. 8x(x + 2)
5. 5mp(5p - 9)
6. 16c(c + 3)
7. y(m + 2n)(m + 4n)
8. -2(x - 8y)(x + 4y)
9. -6(7x - 2y)(x - 7y)
10. -v(m + 2n)(m + 6n)
11. 2(y + 6)(y + 8)
12. 5k²(b + 10)(b - 5)