Precalculus Examples

$\frac{x^{3} + 7 x^{2} - 9 x - 63}{x + 3}$

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.

$\frac{(x^{3} + 7 x^{2}) - 9 x - 63}{x + 3}$

Step 1.1.2

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

$\frac{x^{2} (x + 7) - 9 (x + 7)}{x + 3}$

Step 1.2

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

$\frac{(x + 7) (x^{2} - 9)}{x + 3}$

Step 1.3

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 2

Cancel the common factor of $x + 3$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(x + 7) (x - 3)$ by $1$ .

$(x + 7) (x - 3)$

Step 3

Expand $(x + 7) (x - 3)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$x (x - 3) + 7 (x - 3)$

Step 3.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 7 (x - 3)$

Step 3.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 7 x + 7 \cdot - 3$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 + 7 x + 7 \cdot - 3$

Step 4.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x + 7 x + 7 \cdot - 3$

Step 4.1.3

Multiply $7$ by $- 3$ .

$x^{2} - 3 x + 7 x - 21$

Step 4.2

Add $- 3 x$ and $7 x$ .

$x^{2} + 4 x - 21$