Factoring by Grouping Worksheet - Math Monks - Free Printable

Let's solve each problem by factoring by grouping, step-by-step.

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1. $ x^2 + 2x + 5x + 10 $

First, combine like terms:
$$
x^2 + 7x + 10
$$

Now factor:
We need two numbers that multiply to $10$ and add to $7$: $2$ and $5$

$$
= (x + 2)(x + 5)
$$

✔ Answer: $ (x + 2)(x + 5) $

---

2. $ 49x^3 - 35x^2 + 56x - 40 $

Group into pairs:
$$
(49x^3 - 35x^2) + (56x - 40)
$$

Factor each group:
- $7x^2(7x - 5)$
- $8(7x - 5)$

Now factor out the common binomial:
$$
(7x - 5)(7x^2 + 8)
$$

✔ Answer: $ (7x - 5)(7x^2 + 8) $

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3. $ 28x^4 + 112x^3 - 21x^2 - 84x $

Factor out the GCF first: all terms have $7x$
$$
7x(4x^3 + 16x^2 - 3x - 12)
$$

Now group inside:
$$
7x[(4x^3 + 16x^2) + (-3x - 12)]
$$

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

So:
$$
7x[(x + 4)(4x^2 - 3)]
$$

✔ Answer: $ 7x(x + 4)(4x^2 - 3) $

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

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

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

Common factor:
$$
(x - 2)(x^2 + 5)
$$

✔ Answer: $ (x - 2)(x^2 + 5) $

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5. $ 6x^3 + 3x^2 + 8x + 4 $

Group:
$$
(6x^3 + 3x^2) + (8x + 4)
$$

Factor:
- $3x^2(2x + 1)$
- $4(2x + 1)$

Common factor:
$$
(2x + 1)(3x^2 + 4)
$$

✔ Answer: $ (2x + 1)(3x^2 + 4) $

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6. $ 15xy + 25vy + 30vx + 18x^2 $

Rearrange to group similar terms:
$$
(15xy + 25vy) + (30vx + 18x^2)
$$

But better to look for patterns. Let's try:
Group as:
$$
(15xy + 30vx) + (25vy + 18x^2)
$$

Wait — maybe better to rearrange:
Try:
$$
(15xy + 30vx) + (25vy + 18x^2)
$$ → not helpful.

Alternate idea: group by variables.

Try:
$$
(15xy + 25vy) + (30vx + 18x^2)
$$

Factor:
- $5y(3x + 5v)$
- $6x(5v + 3x)$ → wait, $30vx + 18x^2 = 6x(5v + 3x)$

Note: $3x + 5v$ vs $5v + 3x$ → same!

So:
$$
5y(3x + 5v) + 6x(3x + 5v)
$$

Now factor:
$$
(3x + 5v)(5y + 6x)
$$

✔ Answer: $ (3x + 5v)(5y + 6x) $

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7. $ 5y^2 + 24x + 4y + 30xy $

Rearrange terms to group:
$$
5y^2 + 4y + 30xy + 24x
$$

Group:
$$
(5y^2 + 4y) + (30xy + 24x)
$$

Factor:
- $y(5y + 4)$
- $6x(5y + 4)$

Common factor:
$$
(5y + 4)(y + 6x)
$$

✔ Answer: $ (5y + 4)(y + 6x) $

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8. $ 25v^3 + 5v^2 + 30v + 6 $

Group:
$$
(25v^3 + 5v^2) + (30v + 6)
$$

Factor:
- $5v^2(5v + 1)$
- $6(5v + 1)$

Common factor:
$$
(5v + 1)(5v^2 + 6)
$$

✔ Answer: $ (5v + 1)(5v^2 + 6) $

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

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

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

Common factor:
$$
(x - 1)(x^2 + 2)
$$

✔ Answer: $ (x - 1)(x^2 + 2) $

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10. $ 4xy - 24y - x + 6 $

Group:
$$
(4xy - 24y) + (-x + 6)
$$

Factor:
- $4y(x - 6)$
- $-1(x - 6)$ → or $-(x - 6)$

So:
$$
4y(x - 6) -1(x - 6)
= (x - 6)(4y - 1)
$$

✔ Answer: $ (x - 6)(4y - 1) $

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✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ (x + 2)(x + 5) $ |
| 2 | $ (7x - 5)(7x^2 + 8) $ |
| 3 | $ 7x(x + 4)(4x^2 - 3) $ |
| 4 | $ (x - 2)(x^2 + 5) $ |
| 5 | $ (2x + 1)(3x^2 + 4) $ |
| 6 | $ (3x + 5v)(5y + 6x) $ |
| 7 | $ (5y + 4)(y + 6x) $ |
| 8 | $ (5v + 1)(5v^2 + 6) $ |
| 9 | $ (x - 1)(x^2 + 2) $ |
| 10 | $ (x - 6)(4y - 1) $ |

Let me know if you'd like explanations for any specific step!