Algebra Examples

$\frac{x^{3} + 7 x^{2} - 4 x - 28}{2 x^{2} - 3 x - 2}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$\frac{(x^{3} + 7 x^{2}) - 4 x - 28}{2 x^{2} - 3 x - 2}$

Step 1.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x^{2} (x + 7) - 4 (x + 7)}{2 x^{2} - 3 x - 2}$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $x + 7$ .

$\frac{(x + 7) (x^{2} - 4)}{2 x^{2} - 3 x - 2}$

Step 1.3

Rewrite $4$ as $2^{2}$ .

$\frac{(x + 7) (x^{2} - 2^{2})}{2 x^{2} - 3 x - 2}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{(x + 7) (x + 2) (x - 2)}{2 x^{2} - 3 x - 2}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 2.1.1

Factor $- 3$ out of $- 3 x$ .

$\frac{(x + 7) (x + 2) (x - 2)}{2 x^{2} - 3 (x) - 2}$

Step 2.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$\frac{(x + 7) (x + 2) (x - 2)}{2 x^{2} + (1 - 4) x - 2}$

Step 2.1.3

Apply the distributive property.

$\frac{(x + 7) (x + 2) (x - 2)}{2 x^{2} + 1 x - 4 x - 2}$

Step 2.1.4

Multiply $x$ by $1$ .

$\frac{(x + 7) (x + 2) (x - 2)}{2 x^{2} + x - 4 x - 2}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(x + 7) (x + 2) (x - 2)}{(2 x^{2} + x) - 4 x - 2}$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(x + 7) (x + 2) (x - 2)}{x (2 x + 1) - 2 (2 x + 1)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(x + 7) (x + 2) (x - 2)}{(2 x + 1) (x - 2)}$

Step 3

Cancel the common factor of $x - 2$ .

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Step 3.1

Cancel the common factor.

$\frac{(x + 7) (x + 2) (x - 2)}{(2 x + 1) (x - 2)}$

Step 3.2

Rewrite the expression.

$\frac{(x + 7) (x + 2)}{2 x + 1}$