Multiplying Polynomials Worksheets - Math Monks - Free Printable

Let's solve each problem step by step. The task is to multiply a monomial by a polynomial using the distributive property:

> $$
> a(b + c + d) = ab + ac + ad
> $$

We will distribute the monomial (the single term outside the parentheses) to each term inside the polynomial.

---

1. $ 3q(q^2 - 6q + 5) $

Distribute $ 3q $ to each term:
- $ 3q \cdot q^2 = 3q^3 $
- $ 3q \cdot (-6q) = -18q^2 $
- $ 3q \cdot 5 = 15q $

✔ Answer: $ 3q^3 - 18q^2 + 15q $

---

2. $ 9(x^2 + xy - 8y^2) $

Distribute $ 9 $:
- $ 9 \cdot x^2 = 9x^2 $
- $ 9 \cdot xy = 9xy $
- $ 9 \cdot (-8y^2) = -72y^2 $

✔ Answer: $ 9x^2 + 9xy - 72y^2 $

---

3. $ 7(6x^2 + 9xy + 10y^2) $

Distribute $ 7 $:
- $ 7 \cdot 6x^2 = 42x^2 $
- $ 7 \cdot 9xy = 63xy $
- $ 7 \cdot 10y^2 = 70y^2 $

✔ Answer: $ 42x^2 + 63xy + 70y^2 $

---

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

Distribute $ 3x $:
- $ 3x \cdot 2x^2 = 6x^3 $
- $ 3x \cdot (-5x) = -15x^2 $
- $ 3x \cdot 8 = 24x $

✔ Answer: $ 6x^3 - 15x^2 + 24x $

---

5. $ x^2(-x^2 + 2x + 7) $

Distribute $ x^2 $:
- $ x^2 \cdot (-x^2) = -x^4 $
- $ x^2 \cdot 2x = 2x^3 $
- $ x^2 \cdot 7 = 7x^2 $

✔ Answer: $ -x^4 + 2x^3 + 7x^2 $

---

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

Distribute $ 10 $:
- $ 10 \cdot (-x^2) = -10x^2 $
- $ 10 \cdot 10x = 100x $
- $ 10 \cdot (-6) = -60 $

✔ Answer: $ -10x^2 + 100x - 60 $

---

7. $ 2k^2(2k^3 + 6k - 4) $

Distribute $ 2k^2 $:
- $ 2k^2 \cdot 2k^3 = 4k^5 $
- $ 2k^2 \cdot 6k = 12k^3 $
- $ 2k^2 \cdot (-4) = -8k^2 $

✔ Answer: $ 4k^5 + 12k^3 - 8k^2 $

---

8. $ 6x^2(3x^2 - 5xy - 6) $

Distribute $ 6x^2 $:
- $ 6x^2 \cdot 3x^2 = 18x^4 $
- $ 6x^2 \cdot (-5xy) = -30x^3y $
- $ 6x^2 \cdot (-6) = -36x^2 $

✔ Answer: $ 18x^4 - 30x^3y - 36x^2 $

---

9. $ -x(-x^2 - 4x + 4xy) $

Distribute $ -x $:
- $ -x \cdot (-x^2) = x^3 $
- $ -x \cdot (-4x) = 4x^2 $
- $ -x \cdot 4xy = -4x^2y $

✔ Answer: $ x^3 + 4x^2 - 4x^2y $

---

10. $ 3r(2r^2 - 3ry + 2r) $

Distribute $ 3r $:
- $ 3r \cdot 2r^2 = 6r^3 $
- $ 3r \cdot (-3ry) = -9r^2y $
- $ 3r \cdot 2r = 6r^2 $

✔ Answer: $ 6r^3 - 9r^2y + 6r^2 $

---

✔ Final Answers:

1. $ 3q^3 - 18q^2 + 15q $
2. $ 9x^2 + 9xy - 72y^2 $
3. $ 42x^2 + 63xy + 70y^2 $
4. $ 6x^3 - 15x^2 + 24x $
5. $ -x^4 + 2x^3 + 7x^2 $
6. $ -10x^2 + 100x - 60 $
7. $ 4k^5 + 12k^3 - 8k^2 $
8. $ 18x^4 - 30x^3y - 36x^2 $
9. $ x^3 + 4x^2 - 4x^2y $
10. $ 6r^3 - 9r^2y + 6r^2 $

Let me know if you'd like these written in a formatted worksheet or need explanations for any specific step!