Precalculus Examples

$\frac{\frac{x^{3}}{x + 7}}{\frac{x}{x^{2} + 14 x + 49}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{3}}{x + 7} \cdot \frac{x^{2} + 14 x + 49}{x}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2}}{x + 7} \cdot \frac{x^{2} + 14 x + 49}{x}$

Step 2.1.2

Cancel the common factor.

$\frac{x \cdot x^{2}}{x + 7} \cdot \frac{x^{2} + 14 x + 49}{x}$

Step 2.1.3

Rewrite the expression.

$\frac{x^{2}}{x + 7} (x^{2} + 14 x + 49)$

Step 2.2

Multiply $\frac{x^{2}}{x + 7}$ by $x^{2} + 14 x + 49$ .

$\frac{x^{2} (x^{2} + 14 x + 49)}{x + 7}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{x^{2} (x^{2} + 14 x + 7^{2})}{x + 7}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of ${(x + 7)}^{2}$ and $x + 7$ .

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Step 4.1.1

Factor $x + 7$ out of $x^{2} {(x + 7)}^{2}$ .

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Multiply by $1$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

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

Step 4.1.2.4

Divide $x^{2} (x + 7)$ by $1$ .

$x^{2} (x + 7)$

Step 4.2

Apply the distributive property.

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

Step 5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.1

Multiply $x^{2}$ by $x$ .

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Step 5.1.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} + x^{2} \cdot 7$

Step 5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot 7$

Step 5.2

Add $2$ and $1$ .

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

Step 6

Move $7$ to the left of $x^{2}$ .

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