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:
\[
3(x + 4) = 3 \cdot x + 3 \cdot 4
\]
- Simplify:
\[
3x + 12
\]

Answer:
\[
\boxed{3x + 12}
\]

---

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

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

Answer:
\[
\boxed{8x - 56}
\]

---

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

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

Answer:
\[
\boxed{-2x + 6}
\]

---

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

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

Answer:
\[
\boxed{10a - 20b}
\]

---

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

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

Answer:
\[
\boxed{7x^2 + 28x - 21}
\]

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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:
\[
c^2d^3(5cd^7 - 3c^3d^2 - 4d^3) = c^2d^3 \cdot 5cd^7 + c^2d^3 \cdot (-3c^3d^2) + c^2d^3 \cdot (-4d^3)
\]
- Simplify each term:
\[
c^2d^3 \cdot 5cd^7 = 5c^{2+1}d^{3+7} = 5c^3d^{10}
\]
\[
c^2d^3 \cdot (-3c^3d^2) = -3c^{2+3}d^{3+2} = -3c^5d^5
\]
\[
c^2d^3 \cdot (-4d^3) = -4c^2d^{3+3} = -4c^2d^6
\]
- Combine the terms:
\[
5c^3d^{10} - 3c^5d^5 - 4c^2d^6
\]

Answer:
\[
\boxed{5c^3d^{10} - 3c^5d^5 - 4c^2d^6}
\]

---

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

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

Answer:
\[
\boxed{15x^2 - 35x}
\]

---

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

- First, distribute the \( 3 \) in the left-hand side:
\[
3(y - 2) = 3 \cdot y + 3 \cdot (-2) = 3y - 6
\]
- Substitute back into the equation:
\[
3y - 6 + 2y = 4y + 14
\]
- Combine like terms on the left-hand 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:
\[
\boxed{y = 20}
\]

---

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

- Distribute the monomial \( 2x^2y \) to each term inside the parentheses:
\[
2x^2y(3xy - 2y) = 2x^2y \cdot 3xy + 2x^2y \cdot (-2y)
\]
- Simplify each term:
\[
2x^2y \cdot 3xy = 6x^{2+1}y^{1+1} = 6x^3y^2
\]
\[
2x^2y \cdot (-2y) = -4x^2y^{1+1} = -4x^2y^2
\]
- Combine the terms:
\[
6x^3y^2 - 4x^2y^2
\]

Answer:
\[
\boxed{6x^3y^2 - 4x^2y^2}
\]

---

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

- First, distribute \( d \) in both terms on the left-hand side:
\[
d(d + 3) = d \cdot d + d \cdot 3 = d^2 + 3d
\]
\[
d(d - 4) = d \cdot d + d \cdot (-4) = d^2 - 4d
\]
- Substitute back into the equation:
\[
(d^2 + 3d) - (d^2 - 4d) = 9d - 16
\]
- Simplify the left-hand side by distributing the negative sign:
\[
d^2 + 3d - d^2 + 4d = 9d - 16
\]
- Combine like terms:
\[
(d^2 - d^2) + (3d + 4d) = 9d - 16
\]
\[
0 + 7d = 9d - 16
\]
\[
7d = 9d - 16
\]
- Subtract \( 9d \) from both sides:
\[
7d - 9d = -16
\]
\[
-2d = -16
\]
- Divide by \(-2\):
\[
d = 8
\]

Answer:
\[
\boxed{d = 8}
\]

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Final Answers:

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