Polynomial division worksheet with 10 problems for math practice.

Let's solve each of the polynomial division problems step by step. I'll go through each one, showing the division process and simplifying where possible.

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1.) $(24x^4 + 24x^3 + 3x^2) \div 8x^2$

We divide each term in the numerator by $8x^2$:

$$
\frac{24x^4}{8x^2} = 3x^2, \quad \frac{24x^3}{8x^2} = 3x, \quad \frac{3x^2}{8x^2} = \frac{3}{8}
$$

So the result is:
$$
3x^2 + 3x + \frac{3}{8}
$$

✔ Answer: $3x^2 + 3x + \frac{3}{8}$

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2.) $(16x^5 + 2x^4 + x^3) \div 4x$

Divide each term by $4x$:

$$
\frac{16x^5}{4x} = 4x^4, \quad \frac{2x^4}{4x} = \frac{1}{2}x^3, \quad \frac{x^3}{4x} = \frac{1}{4}x^2
$$

✔ Answer: $4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2$

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3.) $(27x^5 + 9x^4 + 9x^3) \div 9x^3$

Divide each term by $9x^3$:

$$
\frac{27x^5}{9x^3} = 3x^2, \quad \frac{9x^4}{9x^3} = x, \quad \frac{9x^3}{9x^3} = 1
$$

✔ Answer: $3x^2 + x + 1$

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4.) $(2x^4 - 8x^3 + 12x^2) \div 4x^2$

Divide each term by $4x^2$:

$$
\frac{2x^4}{4x^2} = \frac{1}{2}x^2, \quad \frac{-8x^3}{4x^2} = -2x, \quad \frac{12x^2}{4x^2} = 3
$$

✔ Answer: $\frac{1}{2}x^2 - 2x + 3$

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5.) $(x^2 + x + 17) \div (x - 4)$

Use polynomial long division.

Divide $x^2 + x + 17$ by $x - 4$:

1. Divide $x^2 \div x = x$
2. Multiply: $x(x - 4) = x^2 - 4x$
3. Subtract: $(x^2 + x + 17) - (x^2 - 4x) = 5x + 17$
4. Divide $5x \div x = 5$
5. Multiply: $5(x - 4) = 5x - 20$
6. Subtract: $(5x + 17) - (5x - 20) = 37$

So quotient is $x + 5$, remainder $37$

✔ Answer: $x + 5 + \frac{37}{x - 4}$

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6.) $(x^2 + x - 16) \div (x - 4)$

Long division:

1. $x^2 \div x = x$
2. $x(x - 4) = x^2 - 4x$
3. Subtract: $(x^2 + x - 16) - (x^2 - 4x) = 5x - 16$
4. $5x \div x = 5$
5. $5(x - 4) = 5x - 20$
6. Subtract: $(5x - 16) - (5x - 20) = 4$

Quotient: $x + 5$, remainder: $4$

✔ Answer: $x + 5 + \frac{4}{x - 4}$

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7.) $(x^2 - 6x + 2) \div (x - 1)$

Long division:

1. $x^2 \div x = x$
2. $x(x - 1) = x^2 - x$
3. Subtract: $(x^2 - 6x + 2) - (x^2 - x) = -5x + 2$
4. $-5x \div x = -5$
5. $-5(x - 1) = -5x + 5$
6. Subtract: $(-5x + 2) - (-5x + 5) = -3$

Quotient: $x - 5$, remainder: $-3$

✔ Answer: $x - 5 - \frac{3}{x - 1}$

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8.) $(x^2 - 24) \div (x - 5)$

Note: $x^2 - 24$ is not a perfect square, but let’s divide:

1. $x^2 \div x = x$
2. $x(x - 5) = x^2 - 5x$
3. Subtract: $(x^2 - 24) - (x^2 - 5x) = 5x - 24$
4. $5x \div x = 5$
5. $5(x - 5) = 5x - 25$
6. Subtract: $(5x - 24) - (5x - 25) = 1$

Quotient: $x + 5$, remainder: $1$

✔ Answer: $x + 5 + \frac{1}{x - 5}$

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9.) $(2x^2 - 8x + 2) \div (x - 3)$

Long division:

1. $2x^2 \div x = 2x$
2. $2x(x - 3) = 2x^2 - 6x$
3. Subtract: $(2x^2 - 8x + 2) - (2x^2 - 6x) = -2x + 2$
4. $-2x \div x = -2$
5. $-2(x - 3) = -2x + 6$
6. Subtract: $(-2x + 2) - (-2x + 6) = -4$

Quotient: $2x - 2$, remainder: $-4$

✔ Answer: $2x - 2 - \frac{4}{x - 3}$

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10.) $(2x^2 + 5x - 11) \div (x - 4)$

Long division:

1. $2x^2 \div x = 2x$
2. $2x(x - 4) = 2x^2 - 8x$
3. Subtract: $(2x^2 + 5x - 11) - (2x^2 - 8x) = 13x - 11$
4. $13x \div x = 13$
5. $13(x - 4) = 13x - 52$
6. Subtract: $(13x - 11) - (13x - 52) = 41$

Quotient: $2x + 13$, remainder: $41$

✔ Answer: $2x + 13 + \frac{41}{x - 4}$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $3x^2 + 3x + \frac{3}{8}$ |
| 2 | $4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2$ |
| 3 | $3x^2 + x + 1$ |
| 4 | $\frac{1}{2}x^2 - 2x + 3$ |
| 5 | $x + 5 + \frac{37}{x - 4}$ |
| 6 | $x + 5 + \frac{4}{x - 4}$ |
| 7 | $x - 5 - \frac{3}{x - 1}$ |
| 8 | $x + 5 + \frac{1}{x - 5}$ |
| 9 | $2x - 2 - \frac{4}{x - 3}$ |
| 10 | $2x + 13 + \frac{41}{x - 4}$ |

Let me know if you'd like these explained with visual long division steps!