Math worksheet for practicing multiplication of monomials by trinomials.

Let's solve each problem step by step. The task is to multiply a monomial by a trinomial and simplify each expression.

We'll use the distributive property:
$$
a(b + c + d) = ab + ac + ad
$$

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1. $ 6s^5(-3s^4 - 2s^3 + s^2) $

Distribute $6s^5$ to each term:

- $6s^5 \cdot (-3s^4) = -18s^{5+4} = -18s^9$
- $6s^5 \cdot (-2s^3) = -12s^{5+3} = -12s^8$
- $6s^5 \cdot s^2 = 6s^{5+2} = 6s^7$

✔ Answer: $-18s^9 - 12s^8 + 6s^7$

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2. $ 5b^3(4b^5 - 8b^4 + 9b^3) $

Distribute $5b^3$:

- $5b^3 \cdot 4b^5 = 20b^{3+5} = 20b^8$
- $5b^3 \cdot (-8b^4) = -40b^{3+4} = -40b^7$
- $5b^3 \cdot 9b^3 = 45b^{3+3} = 45b^6$

✔ Answer: $20b^8 - 40b^7 + 45b^6$

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3. $ s^4(-s^4 + 9s^3 + 2s^2) $

Distribute $s^4$:

- $s^4 \cdot (-s^4) = -s^{4+4} = -s^8$
- $s^4 \cdot 9s^3 = 9s^{4+3} = 9s^7$
- $s^4 \cdot 2s^2 = 2s^{4+2} = 2s^6$

✔ Answer: $-s^8 + 9s^7 + 2s^6$

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4. $ -5z^4(-z^3 + 3z^2 + 5z) $

Distribute $-5z^4$:

- $-5z^4 \cdot (-z^3) = 5z^{4+3} = 5z^7$
- $-5z^4 \cdot 3z^2 = -15z^{4+2} = -15z^6$
- $-5z^4 \cdot 5z = -25z^{4+1} = -25z^5$

✔ Answer: $5z^7 - 15z^6 - 25z^5$

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

Distribute $-8s^3$:

- $-8s^3 \cdot 2s^3 = -16s^{3+3} = -16s^6$
- $-8s^3 \cdot 4s^2 = -32s^{3+2} = -32s^5$
- $-8s^3 \cdot (-2s) = 16s^{3+1} = 16s^4$

✔ Answer: $-16s^6 - 32s^5 + 16s^4$

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6. $ a^5(-9a^4 - 9a^3 + 4a^2) $

Distribute $a^5$:

- $a^5 \cdot (-9a^4) = -9a^{5+4} = -9a^9$
- $a^5 \cdot (-9a^3) = -9a^{5+3} = -9a^8$
- $a^5 \cdot 4a^2 = 4a^{5+2} = 4a^7$

✔ Answer: $-9a^9 - 9a^8 + 4a^7$

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7. $ 3f^5(-3f^2 - 2f + 1) $

Distribute $3f^5$:

- $3f^5 \cdot (-3f^2) = -9f^{5+2} = -9f^7$
- $3f^5 \cdot (-2f) = -6f^{5+1} = -6f^6$
- $3f^5 \cdot 1 = 3f^5$

✔ Answer: $-9f^7 - 6f^6 + 3f^5$

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8. $ 8p^4(-9p^3 - 7p^2 + 3p) $

Distribute $8p^4$:

- $8p^4 \cdot (-9p^3) = -72p^{4+3} = -72p^7$
- $8p^4 \cdot (-7p^2) = -56p^{4+2} = -56p^6$
- $8p^4 \cdot 3p = 24p^{4+1} = 24p^5$

✔ Answer: $-72p^7 - 56p^6 + 24p^5$

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9. $ -6r^3(7r^3 - 5r^2 + 4r) $

Distribute $-6r^3$:

- $-6r^3 \cdot 7r^3 = -42r^{3+3} = -42r^6$
- $-6r^3 \cdot (-5r^2) = 30r^{3+2} = 30r^5$
- $-6r^3 \cdot 4r = -24r^{3+1} = -24r^4$

✔ Answer: $-42r^6 + 30r^5 - 24r^4$

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10. $ -2n^3(5n^2 + 7n - 6) $

Distribute $-2n^3$:

- $-2n^3 \cdot 5n^2 = -10n^{3+2} = -10n^5$
- $-2n^3 \cdot 7n = -14n^{3+1} = -14n^4$
- $-2n^3 \cdot (-6) = 12n^3$

✔ Answer: $-10n^5 - 14n^4 + 12n^3$

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✔ Final Answers:

1. $-18s^9 - 12s^8 + 6s^7$
2. $20b^8 - 40b^7 + 45b^6$
3. $-s^8 + 9s^7 + 2s^6$
4. $5z^7 - 15z^6 - 25z^5$
5. $-16s^6 - 32s^5 + 16s^4$
6. $-9a^9 - 9a^8 + 4a^7$
7. $-9f^7 - 6f^6 + 3f^5$
8. $-72p^7 - 56p^6 + 24p^5$
9. $-42r^6 + 30r^5 - 24r^4$
10. $-10n^5 - 14n^4 + 12n^3$

Let me know if you'd like these formatted differently or need help understanding any step!