Dividing Polynomials Worksheets - Free Printable

Let's solve each of the monomial division problems step by step. The goal is to divide one monomial by another, simplifying using the rules of exponents.

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Rules to Remember:

- When dividing powers with the same base:
$$
\frac{a^m}{a^n} = a^{m-n}
$$
- Coefficients (numbers) are divided normally.
- Variables are simplified using exponent rules.

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Now, let’s solve each problem:

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1) $ \frac{32x^4y^5}{-16x^2y^3} $

- Divide coefficients: $ \frac{32}{-16} = -2 $
- Divide variables:
- $ x^4 / x^2 = x^{4-2} = x^2 $
- $ y^5 / y^3 = y^{5-3} = y^2 $
- Final answer: $ -2x^2y^2 $

✔ Answer: $ -2x^2y^2 $

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2) $ \frac{-8xy^2}{(-2xy)^2} $

First, simplify the denominator:
- $ (-2xy)^2 = (-2)^2 \cdot x^2 \cdot y^2 = 4x^2y^2 $

Now divide:
- $ \frac{-8xy^2}{4x^2y^2} $
- Coefficients: $ \frac{-8}{4} = -2 $
- $ x / x^2 = x^{1-2} = x^{-1} = \frac{1}{x} $
- $ y^2 / y^2 = y^{0} = 1 $
- So result: $ -2 \cdot \frac{1}{x} = -\frac{2}{x} $

✔ Answer: $ -\frac{2}{x} $

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3) $ \frac{-32m^3n^2}{-4n^2} $

- Coefficients: $ \frac{-32}{-4} = 8 $
- $ m^3 $ stays (no m in denominator)
- $ n^2 / n^2 = n^{0} = 1 $
- So result: $ 8m^3 $

✔ Answer: $ 8m^3 $

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4) $ \frac{pq^2r^4}{-q^3r^2} $

- Coefficients: $ \frac{1}{-1} = -1 $
- $ p $: no match → stays
- $ q^2 / q^3 = q^{2-3} = q^{-1} = \frac{1}{q} $
- $ r^4 / r^2 = r^{4-2} = r^2 $
- Combine: $ -p \cdot \frac{1}{q} \cdot r^2 = -\frac{pr^2}{q} $

✔ Answer: $ -\frac{pr^2}{q} $

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5) $ \frac{60x^7y^6}{15x^2y^6} $

- Coefficients: $ \frac{60}{15} = 4 $
- $ x^7 / x^2 = x^{5} $
- $ y^6 / y^6 = y^0 = 1 $
- Result: $ 4x^5 $

✔ Answer: $ 4x^5 $

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6) $ \frac{16a^7b^5}{-24a^4b^3} $

- Coefficients: $ \frac{16}{-24} = -\frac{2}{3} $
- $ a^7 / a^4 = a^{3} $
- $ b^5 / b^3 = b^{2} $
- Result: $ -\frac{2}{3}a^3b^2 $

✔ Answer: $ -\frac{2}{3}a^3b^2 $

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7) $ \frac{54pq^2}{-9q^2} $

- Coefficients: $ \frac{54}{-9} = -6 $
- $ p $: no match → stays
- $ q^2 / q^2 = 1 $
- Result: $ -6p $

✔ Answer: $ -6p $

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8) $ \frac{x^3}{x} \div (-4x) $

First, simplify $ \frac{x^3}{x} = x^{3-1} = x^2 $

Now divide by $ -4x $:
$ \frac{x^2}{-4x} = -\frac{1}{4} \cdot x^{2-1} = -\frac{1}{4}x $

✔ Answer: $ -\frac{1}{4}x $

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9) $ \frac{70a^5b^4}{-7a^2} $

- Coefficients: $ \frac{70}{-7} = -10 $
- $ a^5 / a^2 = a^3 $
- $ b^4 $: no match → stays
- Result: $ -10a^3b^4 $

✔ Answer: $ -10a^3b^4 $

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10) $ \frac{42p^2q^4}{-6p^3q^3} $

- Coefficients: $ \frac{42}{-6} = -7 $
- $ p^2 / p^3 = p^{-1} = \frac{1}{p} $
- $ q^4 / q^3 = q^{1} = q $
- Combine: $ -7 \cdot \frac{1}{p} \cdot q = -\frac{7q}{p} $

✔ Answer: $ -\frac{7q}{p} $

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✔ Final Answers:

1) $ -2x^2y^2 $
2) $ -\frac{2}{x} $
3) $ 8m^3 $
4) $ -\frac{pr^2}{q} $
5) $ 4x^5 $
6) $ -\frac{2}{3}a^3b^2 $
7) $ -6p $
8) $ -\frac{1}{4}x $
9) $ -10a^3b^4 $
10) $ -\frac{7q}{p} $

Let me know if you'd like these formatted differently or need explanations for any specific step!