Factoring by Grouping Worksheet - Math Monks - Free Printable

Let's solve each polynomial by factoring by grouping. This method is typically used for polynomials with four or more terms, where we group terms in pairs and factor out the greatest common factor (GCF) from each pair.

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1. $ x^3 + 3x^2 + 2x + 6 $

Group terms:
$$
(x^3 + 3x^2) + (2x + 6)
$$

Factor GCF from each group:
$$
x^2(x + 3) + 2(x + 3)
$$

Now factor out the common binomial $(x + 3)$:
$$
(x + 3)(x^2 + 2)
$$

✔ Answer: $ \boxed{(x + 3)(x^2 + 2)} $

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2. $ 2x^3 + 6x^2 - 3x - 9 $

Group:
$$
(2x^3 + 6x^2) + (-3x - 9)
$$

Factor GCF:
$$
2x^2(x + 3) - 3(x + 3)
$$

Factor out common binomial $(x + 3)$:
$$
(x + 3)(2x^2 - 3)
$$

✔ Answer: $ \boxed{(x + 3)(2x^2 - 3)} $

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3. $ 12x^3 - 21x^2 + 28x - 49 $

Group:
$$
(12x^3 - 21x^2) + (28x - 49)
$$

Factor GCF:
$$
3x^2(4x - 7) + 7(4x - 7)
$$

Factor out common binomial $(4x - 7)$:
$$
(4x - 7)(3x^2 + 7)
$$

✔ Answer: $ \boxed{(4x - 7)(3x^2 + 7)} $

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4. $ 6x^3 + 4x^2 - 15x - 10 $

Group:
$$
(6x^3 + 4x^2) + (-15x - 10)
$$

Factor GCF:
$$
2x^2(3x + 2) - 5(3x + 2)
$$

Factor out common binomial $(3x + 2)$:
$$
(3x + 2)(2x^2 - 5)
$$

✔ Answer: $ \boxed{(3x + 2)(2x^2 - 5)} $

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5. $ 105x^2 + 175x^2 - 75x - 125 $

Wait! There’s a typo here: two $x^2$ terms? Let's check:

It says: $105x^2 + 175x^2 - 75x - 125$

That’s actually:
$$
(105x^2 + 175x^2) = 280x^2
$$

So it should be:
$$
280x^2 - 75x - 125
$$

But this is a trinomial — not ideal for grouping unless we rewrite it properly.

Wait — perhaps it was meant to be:
$$
105x^3 + 175x^2 - 75x - 125
$$

Because otherwise, it's just a quadratic with no grouping possible.

Assuming a typo and the correct expression is:
$$
105x^3 + 175x^2 - 75x - 125
$$

Now group:
$$
(105x^3 + 175x^2) + (-75x - 125)
$$

Factor GCF:
$$
35x^2(3x + 5) - 25(3x + 5)
$$

Factor out $(3x + 5)$:
$$
(3x + 5)(35x^2 - 25)
$$

Now factor out GCF from second term:
$$
(3x + 5)(5)(7x^2 - 5) = 5(3x + 5)(7x^2 - 5)
$$

✔ Answer: $ \boxed{5(3x + 5)(7x^2 - 5)} $

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6. $ 21x^3 - 84x^2 + 15x - 60 $

Group:
$$
(21x^3 - 84x^2) + (15x - 60)
$$

Factor GCF:
$$
21x^2(x - 4) + 15(x - 4)
$$

Factor out $(x - 4)$:
$$
(x - 4)(21x^2 + 15)
$$

Factor GCF from second term:
$$
(x - 4)(3)(7x^2 + 5) = 3(x - 4)(7x^2 + 5)
$$

✔ Answer: $ \boxed{3(x - 4)(7x^2 + 5)} $

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7. $ x^3 - 2x^2 + 5x - 10 $

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

Factor GCF:
$$
x^2(x - 2) + 5(x - 2)
$$

Factor out $(x - 2)$:
$$
(x - 2)(x^2 + 5)
$$

✔ Answer: $ \boxed{(x - 2)(x^2 + 5)} $

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8. $ 49x^3 + 35x^2 - 42x - 30 $

Group:
$$
(49x^3 + 35x^2) + (-42x - 30)
$$

Factor GCF:
$$
7x^2(7x + 5) - 6(7x + 5)
$$

Factor out $(7x + 5)$:
$$
(7x + 5)(7x^2 - 6)
$$

✔ Answer: $ \boxed{(7x + 5)(7x^2 - 6)} $

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✔ Final Answers:

1. $ (x + 3)(x^2 + 2) $
2. $ (x + 3)(2x^2 - 3) $
3. $ (4x - 7)(3x^2 + 7) $
4. $ (3x + 2)(2x^2 - 5) $
5. $ 5(3x + 5)(7x^2 - 5) $
6. $ 3(x - 4)(7x^2 + 5) $
7. $ (x - 2)(x^2 + 5) $
8. $ (7x + 5)(7x^2 - 6) $

> ⚠️ Note: Problem #5 had a likely typo (two $x^2$ terms). I assumed it was $105x^3 + 175x^2 - 75x - 125$. If not, please clarify.

Let me know if you'd like these checked using another method or want explanations on factoring steps!