Divide Polynomials Worksheet-1 Worksheets - Free Printable

Problem: Divide the Polynomials

The task involves dividing polynomials using various methods, such as long division, synthetic division, or factoring. Below, I will solve each problem step by step.

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

#### Solution:
1. Factor out the common term in the numerator:
$$
24x^4 + 24x^3 + 3x^2 = 3x^2(8x^2 + 8x + 1)
$$
2. Divide by $8x^2$:
$$
\frac{24x^4 + 24x^3 + 3x^2}{8x^2} = \frac{3x^2(8x^2 + 8x + 1)}{8x^2}
$$
3. Simplify:
$$
\frac{3x^2}{8x^2} \cdot (8x^2 + 8x + 1) = \frac{3}{8}(8x^2 + 8x + 1)
$$
4. Distribute $\frac{3}{8}$:
$$
\frac{3}{8} \cdot 8x^2 + \frac{3}{8} \cdot 8x + \frac{3}{8} \cdot 1 = 3x^2 + 3x + \frac{3}{8}
$$

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

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

#### Solution:
1. Divide each term in the numerator by $4x$:
$$
\frac{16x^5}{4x} + \frac{2x^4}{4x} - \frac{x^3}{4x}
$$
2. Simplify each term:
$$
\frac{16x^5}{4x} = 4x^4, \quad \frac{2x^4}{4x} = \frac{1}{2}x^3, \quad \frac{x^3}{4x} = \frac{1}{4}x^2
$$
3. Combine the results:
$$
4x^4 + \frac{1}{2}x^3 - \frac{1}{4}x^2
$$

#### Final Answer:
$$
\boxed{4x^4 + \frac{1}{2}x^3 - \frac{1}{4}x^2}
$$

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3. $(27x^3 + 9x^2 + 9x) \div 9x$

#### Solution:
1. Divide each term in the numerator by $9x$:
$$
\frac{27x^3}{9x} + \frac{9x^2}{9x} + \frac{9x}{9x}
$$
2. Simplify each term:
$$
\frac{27x^3}{9x} = 3x^2, \quad \frac{9x^2}{9x} = x, \quad \frac{9x}{9x} = 1
$$
3. Combine the results:
$$
3x^2 + x + 1
$$

#### Final Answer:
$$
\boxed{3x^2 + x + 1}
$$

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

#### Solution:
1. Combine like terms in the numerator:
$$
2x^3 + 8x^2 + 12x^2 = 2x^3 + 20x^2
$$
2. Divide each term in the numerator by $4x^2$:
$$
\frac{2x^3}{4x^2} + \frac{20x^2}{4x^2}
$$
3. Simplify each term:
$$
\frac{2x^3}{4x^2} = \frac{1}{2}x, \quad \frac{20x^2}{4x^2} = 5
$$
4. Combine the results:
$$
\frac{1}{2}x + 5
$$

#### Final Answer:
$$
\boxed{\frac{1}{2}x + 5}
$$

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5. $(x^2 + 9x + 17) \div (x + 4)$

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($x^2$) by the leading term of the denominator ($x$):
$$
\frac{x^2}{x} = x
$$
- Multiply $x$ by $(x + 4)$:
$$
x(x + 4) = x^2 + 4x
$$
- Subtract this from the original polynomial:
$$
(x^2 + 9x + 17) - (x^2 + 4x) = 5x + 17
$$
- Divide the leading term of the new polynomial ($5x$) by the leading term of the denominator ($x$):
$$
\frac{5x}{x} = 5
$$
- Multiply $5$ by $(x + 4)$:
$$
5(x + 4) = 5x + 20
$$
- Subtract this from the new polynomial:
$$
(5x + 17) - (5x + 20) = -3
$$
- The quotient is $x + 5$, and the remainder is $-3$.

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

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

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($x^2$) by the leading term of the denominator ($x$):
$$
\frac{x^2}{x} = x
$$
- Multiply $x$ by $(x - 4)$:
$$
x(x - 4) = x^2 - 4x
$$
- Subtract this from the original polynomial:
$$
(x^2 - x - 16) - (x^2 - 4x) = 3x - 16
$$
- Divide the leading term of the new polynomial ($3x$) by the leading term of the denominator ($x$):
$$
\frac{3x}{x} = 3
$$
- Multiply $3$ by $(x - 4)$:
$$
3(x - 4) = 3x - 12
$$
- Subtract this from the new polynomial:
$$
(3x - 16) - (3x - 12) = -4
$$
- The quotient is $x + 3$, and the remainder is $-4$.

#### Final Answer:
$$
\boxed{x + 3 - \frac{4}{x - 4}}
$$

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

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($x^2$) by the leading term of the denominator ($x$):
$$
\frac{x^2}{x} = x
$$
- Multiply $x$ by $(x - 1)$:
$$
x(x - 1) = x^2 - x
$$
- Subtract this from the original polynomial:
$$
(x^2 - 6x + 2) - (x^2 - x) = -5x + 2
$$
- Divide the leading term of the new polynomial ($-5x$) by the leading term of the denominator ($x$):
$$
\frac{-5x}{x} = -5
$$
- Multiply $-5$ by $(x - 1)$:
$$
-5(x - 1) = -5x + 5
$$
- Subtract this from the new polynomial:
$$
(-5x + 2) - (-5x + 5) = -3
$$
- The quotient is $x - 5$, and the remainder is $-3$.

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

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

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($x^2$) by the leading term of the denominator ($x$):
$$
\frac{x^2}{x} = x
$$
- Multiply $x$ by $(x - 5)$:
$$
x(x - 5) = x^2 - 5x
$$
- Subtract this from the original polynomial:
$$
(x^2 - x - 24) - (x^2 - 5x) = 4x - 24
$$
- Divide the leading term of the new polynomial ($4x$) by the leading term of the denominator ($x$):
$$
\frac{4x}{x} = 4
$$
- Multiply $4$ by $(x - 5)$:
$$
4(x - 5) = 4x - 20
$$
- Subtract this from the new polynomial:
$$
(4x - 24) - (4x - 20) = -4
$$
- The quotient is $x + 4$, and the remainder is $-4$.

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

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

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($2x^2$) by the leading term of the denominator ($x$):
$$
\frac{2x^2}{x} = 2x
$$
- Multiply $2x$ by $(x - 3)$:
$$
2x(x - 3) = 2x^2 - 6x
$$
- Subtract this from the original polynomial:
$$
(2x^2 - 8x + 2) - (2x^2 - 6x) = -2x + 2
$$
- Divide the leading term of the new polynomial ($-2x$) by the leading term of the denominator ($x$):
$$
\frac{-2x}{x} = -2
$$
- Multiply $-2$ by $(x - 3)$:
$$
-2(x - 3) = -2x + 6
$$
- Subtract this from the new polynomial:
$$
(-2x + 2) - (-2x + 6) = -4
$$
- The quotient is $2x - 2$, and the remainder is $-4$.

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

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

#### Solution:
1. Use polynomial long division:
- Divide the leading term of the numerator ($2x^2$) by the leading term of the denominator ($x$):
$$
\frac{2x^2}{x} = 2x
$$
- Multiply $2x$ by $(x - 4)$:
$$
2x(x - 4) = 2x^2 - 8x
$$
- Subtract this from the original polynomial:
$$
(2x^2 + 5x - 11) - (2x^2 - 8x) = 13x - 11
$$
- Divide the leading term of the new polynomial ($13x$) by the leading term of the denominator ($x$):
$$
\frac{13x}{x} = 13
$$
- Multiply $13$ by $(x - 4)$:
$$
13(x - 4) = 13x - 52
$$
- Subtract this from the new polynomial:
$$
(13x - 11) - (13x - 52) = 41
$$
- The quotient is $2x + 13$, and the remainder is $41$.

#### Final Answer:
$$
\boxed{2x + 13 + \frac{41}{x - 4}}
$$

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Final Answers:

1. $\boxed{3x^2 + 3x + \frac{3}{8}}$
2. $\boxed{4x^4 + \frac{1}{2}x^3 - \frac{1}{4}x^2}$
3. $\boxed{3x^2 + x + 1}$
4. $\boxed{\frac{1}{2}x + 5}$
5. $\boxed{x + 5 - \frac{3}{x + 4}}$
6. $\boxed{x + 3 - \frac{4}{x - 4}}$
7. $\boxed{x - 5 - \frac{3}{x - 1}}$
8. $\boxed{x + 4 - \frac{4}{x - 5}}$
9. $\boxed{2x - 2 - \frac{4}{x - 3}}$
10. $\boxed{2x + 13 + \frac{41}{x - 4}}$