Multiplying Polynomials Worksheets - Free Printable

Let's solve each of the polynomial multiplication problems step by step. The goal is to distribute the monomial outside the parentheses to each term inside the parentheses, then simplify.

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1) $ 3u^2(3u^2 + 5u - 7u^2 + 12) $

First, simplify inside the parentheses:
$$
3u^2 + 5u - 7u^2 + 12 = (3u^2 - 7u^2) + 5u + 12 = -4u^2 + 5u + 12
$$

Now distribute $ 3u^2 $:
$$
3u^2(-4u^2) + 3u^2(5u) + 3u^2(12) = -12u^4 + 15u^3 + 36u^2
$$

✔ Answer: $ -12u^4 + 15u^3 + 36u^2 $

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2) $ -5c^2(-4c^6 + 8c^4 + 10) $

Distribute $ -5c^2 $ to each term:
$$
-5c^2(-4c^6) = 20c^8 \\
-5c^2(8c^4) = -40c^6 \\
-5c^2(10) = -50c^2
$$

✔ Answer: $ 20c^8 - 40c^6 - 50c^2 $

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3) $ 7a^2(-4a - 6a^3) $

Distribute $ 7a^2 $:
$$
7a^2(-4a) = -28a^3 \\
7a^2(-6a^3) = -42a^5
$$

✔ Answer: $ -42a^5 - 28a^3 $

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

Distribute $ 9n $:
$$
9n(-2n^3) = -18n^4 \\
9n(6n^5) = 54n^6 \\
9n(-7n^2) = -63n^3
$$

Now arrange in descending order:
$$
54n^6 - 18n^4 - 63n^3
$$

✔ Answer: $ 54n^6 - 18n^4 - 63n^3 $

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5) $ -x^5(1 + 17x^6) $

Distribute $ -x^5 $:
$$
-x^5(1) = -x^5 \\
-x^5(17x^6) = -17x^{11}
$$

✔ Answer: $ -17x^{11} - x^5 $

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6) $ 8y^3(y^4 + 11) $

Distribute $ 8y^3 $:
$$
8y^3(y^4) = 8y^7 \\
8y^3(11) = 88y^3
$$

✔ Answer: $ 8y^7 + 88y^3 $

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7) $ -b^7(4b^6 + 8b^4 + 2b + 9) $

Distribute $ -b^7 $:
$$
-b^7(4b^6) = -4b^{13} \\
-b^7(8b^4) = -8b^{11} \\
-b^7(2b) = -2b^8 \\
-b^7(9) = -9b^7
$$

✔ Answer: $ -4b^{13} - 8b^{11} - 2b^8 - 9b^7 $

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8) $ 10(7m^2 - 3 - m^3) $

First, rewrite in standard form: $ -m^3 + 7m^2 - 3 $

Now distribute $ 10 $:
$$
10(-m^3) = -10m^3 \\
10(7m^2) = 70m^2 \\
10(-3) = -30
$$

✔ Answer: $ -10m^3 + 70m^2 - 30 $

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✔ Final Answers:

1) $ -12u^4 + 15u^3 + 36u^2 $
2) $ 20c^8 - 40c^6 - 50c^2 $
3) $ -42a^5 - 28a^3 $
4) $ 54n^6 - 18n^4 - 63n^3 $
5) $ -17x^{11} - x^5 $
6) $ 8y^7 + 88y^3 $
7) $ -4b^{13} - 8b^{11} - 2b^8 - 9b^7 $
8) $ -10m^3 + 70m^2 - 30 $

Let me know if you'd like these explained with visual steps or diagrams!