Dividing Polynomials Worksheets - Free Printable

Let's solve each of the polynomial division problems step by step. The goal is to divide each polynomial by the given monomial or polynomial, simplifying as much as possible.

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1) $ (6x^3 + 3x^2 + 9x^4) \div 3x $

Step 1: Rewrite in standard form (descending powers):
$$
(9x^4 + 6x^3 + 3x^2) \div 3x
$$

Step 2: Divide each term by $3x$:
$$
\frac{9x^4}{3x} + \frac{6x^3}{3x} + \frac{3x^2}{3x}
= 3x^3 + 2x^2 + x
$$

✔ Answer: $ 3x^3 + 2x^2 + x $

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2) $ (2k^5 - 8k^2 - 6k^3) \div 2k^2 $

Step 1: Rewrite in standard form:
$$
(2k^5 - 6k^3 - 8k^2) \div 2k^2
$$

Step 2: Divide each term by $2k^2$:
$$
\frac{2k^5}{2k^2} - \frac{6k^3}{2k^2} - \frac{8k^2}{2k^2}
= k^3 - 3k - 4
$$

✔ Answer: $ k^3 - 3k - 4 $

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3) $ (5m^4 + 10m^6 - 15) \div 5 $

Step 1: Rewrite in standard form:
$$
(10m^6 + 5m^4 - 15) \div 5
$$

Step 2: Divide each term by 5:
$$
\frac{10m^6}{5} + \frac{5m^4}{5} - \frac{15}{5}
= 2m^6 + m^4 - 3
$$

✔ Answer: $ 2m^6 + m^4 - 3 $

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4) $ (-8n^8 - n^7) \div (-n^4) $

Step 1: Divide each term by $-n^4$:
$$
\frac{-8n^8}{-n^4} + \frac{-n^7}{-n^4}
= 8n^{8-4} + n^{7-4}
= 8n^4 + n^3
$$

✔ Answer: $ 8n^4 + n^3 $

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5) $ (-4w^7 + 6w^6 + w^3 - 6w^5) \div w^3 $

Step 1: Rewrite in standard form:
$$
(-4w^7 - 6w^5 + 6w^6 + w^3) \div w^3
= (-4w^7 + 6w^6 - 6w^5 + w^3) \div w^3
$$

Step 2: Divide each term by $w^3$:
$$
\frac{-4w^7}{w^3} + \frac{6w^6}{w^3} - \frac{6w^5}{w^3} + \frac{w^3}{w^3}
= -4w^4 + 6w^3 - 6w^2 + 1
$$

✔ Answer: $ -4w^4 + 6w^3 - 6w^2 + 1 $

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6) $ (-7x^5 + 7x^3) \div 7x^4 $

Step 1: Divide each term by $7x^4$:
$$
\frac{-7x^5}{7x^4} + \frac{7x^3}{7x^4}
= -x^{5-4} + x^{3-4}
= -x + x^{-1}
$$

But $x^{-1}$ is $ \frac{1}{x} $, so:
$$
-x + \frac{1}{x}
$$

⚠️ This expression has a negative exponent — it’s not a polynomial. But since the question asks to divide, we can write it as:

✔ Answer: $ -x + \frac{1}{x} $

Alternatively, leave as: $ -x + x^{-1} $

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7) $ (8y^3 + 16y) \div (-8y) $

Step 1: Divide each term by $-8y$:
$$
\frac{8y^3}{-8y} + \frac{16y}{-8y}
= -y^{3-1} - 2
= -y^2 - 2
$$

✔ Answer: $ -y^2 - 2 $

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8) $ (6v^6 - 5v^4 + v^7) \div v $

Step 1: Rewrite in standard form:
$$
(v^7 + 6v^6 - 5v^4) \div v
$$

Step 2: Divide each term by $v$:
$$
\frac{v^7}{v} + \frac{6v^6}{v} - \frac{5v^4}{v}
= v^6 + 6v^5 - 5v^3
$$

✔ Answer: $ v^6 + 6v^5 - 5v^3 $

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✔ Final Answers Summary:

1) $ 3x^3 + 2x^2 + x $
2) $ k^3 - 3k - 4 $
3) $ 2m^6 + m^4 - 3 $
4) $ 8n^4 + n^3 $
5) $ -4w^4 + 6w^3 - 6w^2 + 1 $
6) $ -x + \frac{1}{x} $
7) $ -y^2 - 2 $
8) $ v^6 + 6v^5 - 5v^3 $

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Let me know if you'd like these explained with long division or factoring!