Polynomial division worksheet with ten problems for algebra practice.

Here are the step-by-step solutions for each problem on the worksheet.

1) $(24x^4 + 24x^3 + 3x^2) \div 8x^2$
To solve this, divide each term inside the parentheses by $8x^2$.
* $24x^4 \div 8x^2 = 3x^2$ (because $24/8=3$ and $x^4/x^2=x^2$)
* $24x^3 \div 8x^2 = 3x$ (because $24/8=3$ and $x^3/x^2=x$)
* $3x^2 \div 8x^2 = \frac{3}{8}$ (the $x^2$ terms cancel out)

Answer: $3x^2 + 3x + \frac{3}{8}$

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2) $(16x^3 + 2x^4 + x^2) \div 4x$
First, it helps to rewrite the polynomial in standard order (highest power first): $2x^4 + 16x^3 + x^2$. Now divide each term by $4x$.
* $2x^4 \div 4x = \frac{2}{4}x^3 = \frac{1}{2}x^3$
* $16x^3 \div 4x = 4x^2$
* $x^2 \div 4x = \frac{1}{4}x$

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

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3) $(27x^5 + 9x^4 + 9x^3) \div 9x^2$
Divide each term by $9x^2$.
* $27x^5 \div 9x^2 = 3x^3$
* $9x^4 \div 9x^2 = x^2$
* $9x^3 \div 9x^2 = x$

Answer: $3x^3 + x^2 + x$

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4) $(2x^5 + 8x^4 + 12x^3) \div 4x^3$
Divide each term by $4x^3$.
* $2x^5 \div 4x^3 = \frac{2}{4}x^2 = \frac{1}{2}x^2$
* $8x^4 \div 4x^3 = 2x$
* $12x^3 \div 4x^3 = 3$

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

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5) $(x^2 + 9x + 17) \div (x + 4)$
We use polynomial long division or synthetic division here. Let's use synthetic division with $-4$.
Coefficients: $1, 9, 17$
1. Bring down the $1$.
2. Multiply $1 \times -4 = -4$. Add to $9$ to get $5$.
3. Multiply $5 \times -4 = -20$. Add to $17$ to get $-3$.

The result is $1x + 5$ with a remainder of $-3$.

Answer: $x + 5 - \frac{3}{x+4}$

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6) $(x^2 + x - 16) \div (x - 4)$
Use synthetic division with $4$.
Coefficients: $1, 1, -16$
1. Bring down the $1$.
2. Multiply $1 \times 4 = 4$. Add to $1$ to get $5$.
3. Multiply $5 \times 4 = 20$. Add to $-16$ to get $4$.

The result is $1x + 5$ with a remainder of $4$.

Answer: $x + 5 + \frac{4}{x-4}$

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7) $(x^2 - 6x + 2) \div (x - 1)$
Use synthetic division with $1$.
Coefficients: $1, -6, 2$
1. Bring down the $1$.
2. Multiply $1 \times 1 = 1$. Add to $-6$ to get $-5$.
3. Multiply $-5 \times 1 = -5$. Add to $2$ to get $-3$.

The result is $1x - 5$ with a remainder of $-3$.

Answer: $x - 5 - \frac{3}{x-1}$

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8) $(x^2 - x - 24) \div (x - 5)$
Use synthetic division with $5$.
Coefficients: $1, -1, -24$
1. Bring down the $1$.
2. Multiply $1 \times 5 = 5$. Add to $-1$ to get $4$.
3. Multiply $4 \times 5 = 20$. Add to $-24$ to get $-4$.

The result is $1x + 4$ with a remainder of $-4$.

Answer: $x + 4 - \frac{4}{x-5}$

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9) $(2x^2 - 8x + 2) \div (x - 3)$
Use synthetic division with $3$.
Coefficients: $2, -8, 2$
1. Bring down the $2$.
2. Multiply $2 \times 3 = 6$. Add to $-8$ to get $-2$.
3. Multiply $-2 \times 3 = -6$. Add to $2$ to get $-4$.

The result is $2x - 2$ with a remainder of $-4$.

Answer: $2x - 2 - \frac{4}{x-3}$

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10) $(2x^2 + 5x - 11) \div (x + 4)$
Use synthetic division with $-4$.
Coefficients: $2, 5, -11$
1. Bring down the $2$.
2. Multiply $2 \times -4 = -8$. Add to $5$ to get $-3$.
3. Multiply $-3 \times -4 = 12$. Add to $-11$ to get $1$.

The result is $2x - 3$ with a remainder of $1$.

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

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Final Answer:
1) $3x^2 + 3x + \frac{3}{8}$
2) $\frac{1}{2}x^3 + 4x^2 + \frac{1}{4}x$
3) $3x^3 + x^2 + x$
4) $\frac{1}{2}x^2 + 2x + 3$
5) $x + 5 - \frac{3}{x+4}$
6) $x + 5 + \frac{4}{x-4}$
7) $x - 5 - \frac{3}{x-1}$
8) $x + 4 - \frac{4}{x-5}$
9) $2x - 2 - \frac{4}{x-3}$
10) $2x - 3 + \frac{1}{x+4}$