Multiplying Polynomials Worksheets - Math Monks - Free Printable

Let's solve each polynomial multiplication problem step by step using the distributive property (also known as the FOIL method for binomials).

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1. $(a + 4)(a + 7)$

Use FOIL:
- F: $a \cdot a = a^2$
- O: $a \cdot 7 = 7a$
- I: $4 \cdot a = 4a$
- L: $4 \cdot 7 = 28$

Add:
$a^2 + 7a + 4a + 28 = a^2 + 11a + 28$

✔ Answer: $a^2 + 11a + 28$

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2. $(a - 7)(a - 3)$

FOIL:
- F: $a \cdot a = a^2$
- O: $a \cdot (-3) = -3a$
- I: $-7 \cdot a = -7a$
- L: $-7 \cdot (-3) = 21$

Add:
$a^2 - 3a - 7a + 21 = a^2 - 10a + 21$

✔ Answer: $a^2 - 10a + 21$

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3. $(a - 7)(a - 5)$

FOIL:
- F: $a \cdot a = a^2$
- O: $a \cdot (-5) = -5a$
- I: $-7 \cdot a = -7a$
- L: $-7 \cdot (-5) = 35$

Add:
$a^2 - 5a - 7a + 35 = a^2 - 12a + 35$

✔ Answer: $a^2 - 12a + 35$

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4. $(3x - 4)(2x - 8)$

FOIL:
- F: $3x \cdot 2x = 6x^2$
- O: $3x \cdot (-8) = -24x$
- I: $-4 \cdot 2x = -8x$
- L: $-4 \cdot (-8) = 32$

Add:
$6x^2 - 24x - 8x + 32 = 6x^2 - 32x + 32$

✔ Answer: $6x^2 - 32x + 32$

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5. $(x - 2)(x + 1)$

FOIL:
- F: $x \cdot x = x^2$
- O: $x \cdot 1 = x$
- I: $-2 \cdot x = -2x$
- L: $-2 \cdot 1 = -2$

Add:
$x^2 + x - 2x - 2 = x^2 - x - 2$

✔ Answer: $x^2 - x - 2$

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6. $(x - 1)(x^2 + 6x - 1)$

Distribute each term in the first polynomial:

- $x \cdot (x^2 + 6x - 1) = x^3 + 6x^2 - x$
- $-1 \cdot (x^2 + 6x - 1) = -x^2 - 6x + 1$

Now add:
$$
x^3 + 6x^2 - x - x^2 - 6x + 1 = x^3 + (6x^2 - x^2) + (-x - 6x) + 1 = x^3 + 5x^2 - 7x + 1
$$

✔ Answer: $x^3 + 5x^2 - 7x + 1$

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7. $(x - 8)(x - 2)$

FOIL:
- F: $x \cdot x = x^2$
- O: $x \cdot (-2) = -2x$
- I: $-8 \cdot x = -8x$
- L: $-8 \cdot (-2) = 16$

Add:
$x^2 - 2x - 8x + 16 = x^2 - 10x + 16$

✔ Answer: $x^2 - 10x + 16$

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8. $(x + 6)(x - 3)$

FOIL:
- F: $x \cdot x = x^2$
- O: $x \cdot (-3) = -3x$
- I: $6 \cdot x = 6x$
- L: $6 \cdot (-3) = -18$

Add:
$x^2 - 3x + 6x - 18 = x^2 + 3x - 18$

✔ Answer: $x^2 + 3x - 18$

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9. $(4x + 1)(3x - 4)$

FOIL:
- F: $4x \cdot 3x = 12x^2$
- O: $4x \cdot (-4) = -16x$
- I: $1 \cdot 3x = 3x$
- L: $1 \cdot (-4) = -4$

Add:
$12x^2 - 16x + 3x - 4 = 12x^2 - 13x - 4$

✔ Answer: $12x^2 - 13x - 4$

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10. $(2x - 6)(x + 2)$

FOIL:
- F: $2x \cdot x = 2x^2$
- O: $2x \cdot 2 = 4x$
- I: $-6 \cdot x = -6x$
- L: $-6 \cdot 2 = -12$

Add:
$2x^2 + 4x - 6x - 12 = 2x^2 - 2x - 12$

✔ Answer: $2x^2 - 2x - 12$

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✔ Final Answers:

1. $a^2 + 11a + 28$
2. $a^2 - 10a + 21$
3. $a^2 - 12a + 35$
4. $6x^2 - 32x + 32$
5. $x^2 - x - 2$
6. $x^3 + 5x^2 - 7x + 1$
7. $x^2 - 10x + 16$
8. $x^2 + 3x - 18$
9. $12x^2 - 13x - 4$
10. $2x^2 - 2x - 12$

You can now write these answers in your coloring activity and color them as instructed! 🎨