Multiplying polynomials with monomials worksheet - Free Printable

Problem: Multiplying Polynomials with a Monomial

The task involves multiplying polynomials by monomials. Let's solve each problem step by step.

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#### 1. \( 3(x + 4) \)

- Distribute the monomial \( 3 \) to each term inside the parentheses:
\[
3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12
\]

Answer: \( 3x + 12 \)

---

#### 2. \( 8(x - 7) \)

- Distribute the monomial \( 8 \) to each term inside the parentheses:
\[
8(x - 7) = 8 \cdot x + 8 \cdot (-7) = 8x - 56
\]

Answer: \( 8x - 56 \)

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#### 3. \( -2(x - 3) \)

- Distribute the monomial \( -2 \) to each term inside the parentheses:
\[
-2(x - 3) = -2 \cdot x + (-2) \cdot (-3) = -2x + 6
\]

Answer: \( -2x + 6 \)

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#### 4. \( 5(2a - 4b) \)

- Distribute the monomial \( 5 \) to each term inside the parentheses:
\[
5(2a - 4b) = 5 \cdot 2a + 5 \cdot (-4b) = 10a - 20b
\]

Answer: \( 10a - 20b \)

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#### 5. \( 7(x^2 + 4x - 3) \)

- Distribute the monomial \( 7 \) to each term inside the parentheses:
\[
7(x^2 + 4x - 3) = 7 \cdot x^2 + 7 \cdot 4x + 7 \cdot (-3) = 7x^2 + 28x - 21
\]

Answer: \( 7x^2 + 28x - 21 \)

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#### 6. \( c^2d^3(5cd^7 - 3c^3d^2 - 4d^3) \)

- Distribute the monomial \( c^2d^3 \) to each term inside the parentheses:
\[
c^2d^3(5cd^7) = 5c^3d^{10}
\]
\[
c^2d^3(-3c^3d^2) = -3c^5d^5
\]
\[
c^2d^3(-4d^3) = -4c^2d^6
\]
- Combine the results:
\[
c^2d^3(5cd^7 - 3c^3d^2 - 4d^3) = 5c^3d^{10} - 3c^5d^5 - 4c^2d^6
\]

Answer: \( 5c^3d^{10} - 3c^5d^5 - 4c^2d^6 \)

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#### 7. \( -5x(-3x + 7) \)

- Distribute the monomial \( -5x \) to each term inside the parentheses:
\[
-5x(-3x) = 15x^2
\]
\[
-5x(7) = -35x
\]
- Combine the results:
\[
-5x(-3x + 7) = 15x^2 - 35x
\]

Answer: \( 15x^2 - 35x \)

---

#### 8. \( 3(y - 2) + 2y = 4y + 14 \)

- First, distribute the \( 3 \) in \( 3(y - 2) \):
\[
3(y - 2) = 3y - 6
\]
- Substitute back into the equation:
\[
3y - 6 + 2y = 4y + 14
\]
- Combine like terms on the left side:
\[
(3y + 2y) - 6 = 4y + 14
\]
\[
5y - 6 = 4y + 14
\]
- Subtract \( 4y \) from both sides:
\[
5y - 4y - 6 = 14
\]
\[
y - 6 = 14
\]
- Add 6 to both sides:
\[
y = 20
\]

Answer: \( y = 20 \)

---

#### 9. \( 2x^2y(3xy - 2y) \)

- Distribute the monomial \( 2x^2y \) to each term inside the parentheses:
\[
2x^2y(3xy) = 6x^3y^2
\]
\[
2x^2y(-2y) = -4x^2y^2
\]
- Combine the results:
\[
2x^2y(3xy - 2y) = 6x^3y^2 - 4x^2y^2
\]

Answer: \( 6x^3y^2 - 4x^2y^2 \)

---

#### 10. \( d(d + 3) - d(d - 4) = 9d - 16 \)

- First, expand \( d(d + 3) \):
\[
d(d + 3) = d^2 + 3d
\]
- Next, expand \( d(d - 4) \):
\[
d(d - 4) = d^2 - 4d
\]
- Substitute back into the equation:
\[
(d^2 + 3d) - (d^2 - 4d) = 9d - 16
\]
- Simplify the left side by distributing the negative sign:
\[
d^2 + 3d - d^2 + 4d = 9d - 16
\]
- Combine like terms:
\[
(d^2 - d^2) + (3d + 4d) = 9d - 16
\]
\[
7d = 9d - 16
\]
- Subtract \( 9d \) from both sides:
\[
7d - 9d = -16
\]
\[
-2d = -16
\]
- Divide by \(-2\):
\[
d = 8
\]

Answer: \( d = 8 \)

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 3x + 12 \\
2. & \ 8x - 56 \\
3. & \ -2x + 6 \\
4. & \ 10a - 20b \\
5. & \ 7x^2 + 28x - 21 \\
6. & \ 5c^3d^{10} - 3c^5d^5 - 4c^2d^6 \\
7. & \ 15x^2 - 35x \\
8. & \ y = 20 \\
9. & \ 6x^3y^2 - 4x^2y^2 \\
10. & \ d = 8
\end{aligned}
}
\]