Dividing Polynomials Worksheets - Free Printable

To solve the problems involving dividing polynomials using the factorization method, we will follow these steps:

1. Factorize both the numerator and the denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

Let's go through each problem step by step.

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Problem 1: Divide \( x^2 + 5x + 6 \) by \( x + 2 \)

#### Step 1: Factorize the numerator
The numerator is \( x^2 + 5x + 6 \). We need to factorize it:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]

#### Step 2: Write the division
\[ \frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2} \]

#### Step 3: Cancel the common factor
\[ \frac{(x + 2)(x + 3)}{x + 2} = x + 3 \quad \text{(for } x \neq -2\text{)} \]

#### Final Answer:
\[ \boxed{x + 3} \]

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Problem 2: Divide \( x^2 - 9 \) by \( x - 3 \)

#### Step 1: Factorize the numerator
The numerator is \( x^2 - 9 \), which is a difference of squares:
\[ x^2 - 9 = (x - 3)(x + 3) \]

#### Step 2: Write the division
\[ \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} \]

#### Step 3: Cancel the common factor
\[ \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad \text{(for } x \neq 3\text{)} \]

#### Final Answer:
\[ \boxed{x + 3} \]

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Problem 3: Divide \( 12x^2 - 30x + 18 \) by \( 2x - 3 \)

#### Step 1: Factorize the numerator
The numerator is \( 12x^2 - 30x + 18 \). First, factor out the greatest common factor (GCF), which is 6:
\[ 12x^2 - 30x + 18 = 6(2x^2 - 5x + 3) \]

Next, factorize \( 2x^2 - 5x + 3 \):
\[ 2x^2 - 5x + 3 = (2x - 3)(x - 1) \]

So, the fully factorized numerator is:
\[ 12x^2 - 30x + 18 = 6(2x - 3)(x - 1) \]

#### Step 2: Write the division
\[ \frac{12x^2 - 30x + 18}{2x - 3} = \frac{6(2x - 3)(x - 1)}{2x - 3} \]

#### Step 3: Cancel the common factor
\[ \frac{6(2x - 3)(x - 1)}{2x - 3} = 6(x - 1) \quad \text{(for } x \neq \frac{3}{2}\text{)} \]

#### Final Answer:
\[ \boxed{6(x - 1)} \]

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Problem 4: Divide \( 4x^2 - 36xy + 81y^2 \) by \( 2x - 9y \)

#### Step 1: Factorize the numerator
The numerator is \( 4x^2 - 36xy + 81y^2 \), which is a perfect square trinomial:
\[ 4x^2 - 36xy + 81y^2 = (2x - 9y)^2 \]

#### Step 2: Write the division
\[ \frac{4x^2 - 36xy + 81y^2}{2x - 9y} = \frac{(2x - 9y)^2}{2x - 9y} \]

#### Step 3: Cancel the common factor
\[ \frac{(2x - 9y)^2}{2x - 9y} = 2x - 9y \quad \text{(for } x \neq \frac{9y}{2}\text{)} \]

#### Final Answer:
\[ \boxed{2x - 9y} \]

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Problem 5: Divide \( 2x^3 - 14x^2 + 24x \) by \( 2x \)

#### Step 1: Factorize the numerator
The numerator is \( 2x^3 - 14x^2 + 24x \). Factor out the GCF, which is \( 2x \):
\[ 2x^3 - 14x^2 + 24x = 2x(x^2 - 7x + 12) \]

Next, factorize \( x^2 - 7x + 12 \):
\[ x^2 - 7x + 12 = (x - 3)(x - 4) \]

So, the fully factorized numerator is:
\[ 2x^3 - 14x^2 + 24x = 2x(x - 3)(x - 4) \]

#### Step 2: Write the division
\[ \frac{2x^3 - 14x^2 + 24x}{2x} = \frac{2x(x - 3)(x - 4)}{2x} \]

#### Step 3: Cancel the common factor
\[ \frac{2x(x - 3)(x - 4)}{2x} = (x - 3)(x - 4) \quad \text{(for } x \neq 0\text{)} \]

#### Final Answer:
\[ \boxed{(x - 3)(x - 4)} \]

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Problem 6: Divide \( 3x^2 - 12x + 12 \) by \( x - 2 \)

#### Step 1: Factorize the numerator
The numerator is \( 3x^2 - 12x + 12 \). Factor out the GCF, which is 3:
\[ 3x^2 - 12x + 12 = 3(x^2 - 4x + 4) \]

Next, factorize \( x^2 - 4x + 4 \), which is a perfect square trinomial:
\[ x^2 - 4x + 4 = (x - 2)^2 \]

So, the fully factorized numerator is:
\[ 3x^2 - 12x + 12 = 3(x - 2)^2 \]

#### Step 2: Write the division
\[ \frac{3x^2 - 12x + 12}{x - 2} = \frac{3(x - 2)^2}{x - 2} \]

#### Step 3: Cancel the common factor
\[ \frac{3(x - 2)^2}{x - 2} = 3(x - 2) \quad \text{(for } x \neq 2\text{)} \]

#### Final Answer:
\[ \boxed{3(x - 2)} \]

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Problem 7: Divide \( 2x^3 - 8x^2 + 8x \) by \( 2x \)

#### Step 1: Factorize the numerator
The numerator is \( 2x^3 - 8x^2 + 8x \). Factor out the GCF, which is \( 2x \):
\[ 2x^3 - 8x^2 + 8x = 2x(x^2 - 4x + 4) \]

Next, factorize \( x^2 - 4x + 4 \), which is a perfect square trinomial:
\[ x^2 - 4x + 4 = (x - 2)^2 \]

So, the fully factorized numerator is:
\[ 2x^3 - 8x^2 + 8x = 2x(x - 2)^2 \]

#### Step 2: Write the division
\[ \frac{2x^3 - 8x^2 + 8x}{2x} = \frac{2x(x - 2)^2}{2x} \]

#### Step 3: Cancel the common factor
\[ \frac{2x(x - 2)^2}{2x} = (x - 2)^2 \quad \text{(for } x \neq 0\text{)} \]

#### Final Answer:
\[ \boxed{(x - 2)^2} \]

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Final Answers:

1. \( \boxed{x + 3} \)
2. \( \boxed{x + 3} \)
3. \( \boxed{6(x - 1)} \)
4. \( \boxed{2x - 9y} \)
5. \( \boxed{(x - 3)(x - 4)} \)
6. \( \boxed{3(x - 2)} \)
7. \( \boxed{(x - 2)^2} \)