Algebra Examples

$\frac{x^{2} + 10 x + 16}{x^{2} - x} \cdot \frac{x^{2} + 5 x - 6}{x^{2} + 7 x + 10}$

Step 1

Factor $x^{2} + 10 x + 16$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $10$ .

$2, 8$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 8)}{x^{2} - x} \cdot \frac{x^{2} + 5 x - 6}{x^{2} + 7 x + 10}$

Step 2

Factor $x$ out of $x^{2} - x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

$\frac{(x + 2) (x + 8)}{x \cdot x - x} \cdot \frac{x^{2} + 5 x - 6}{x^{2} + 7 x + 10}$

Step 2.2

Factor $x$ out of $- x$ .

$\frac{(x + 2) (x + 8)}{x \cdot x + x \cdot - 1} \cdot \frac{x^{2} + 5 x - 6}{x^{2} + 7 x + 10}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\frac{(x + 2) (x + 8)}{x (x - 1)} \cdot \frac{x^{2} + 5 x - 6}{x^{2} + 7 x + 10}$

Step 3

Factor $x^{2} + 5 x - 6$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 8)}{x (x - 1)} \cdot \frac{(x - 1) (x + 6)}{x^{2} + 7 x + 10}$

Step 4

Factor $x^{2} + 7 x + 10$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $7$ .

$2, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 8)}{x (x - 1)} \cdot \frac{(x - 1) (x + 6)}{(x + 2) (x + 5)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 2$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(x + 2) (x + 8)}{x (x - 1)} \cdot \frac{(x - 1) (x + 6)}{(x + 2) (x + 5)}$

Step 5.1.2

Rewrite the expression.

$\frac{x + 8}{x (x - 1)} \cdot \frac{(x - 1) (x + 6)}{x + 5}$

Step 5.2

Cancel the common factor of $x - 1$ .

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Step 5.2.1

Factor $x - 1$ out of $x (x - 1)$ .

$\frac{x + 8}{(x - 1) x} \cdot \frac{(x - 1) (x + 6)}{x + 5}$

Step 5.2.2

Cancel the common factor.

$\frac{x + 8}{(x - 1) x} \cdot \frac{(x - 1) (x + 6)}{x + 5}$

Step 5.2.3

Rewrite the expression.

$\frac{x + 8}{x} \cdot \frac{x + 6}{x + 5}$

Step 5.3

Multiply $\frac{x + 8}{x}$ by $\frac{x + 6}{x + 5}$ .

$\frac{(x + 8) (x + 6)}{x (x + 5)}$