Algebra Examples

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x^{3} + 14 x^{2} + 48 x}$

Step 1

Simplify the denominator.

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Step 1.1

Factor $x$ out of $x^{3} + 14 x^{2} + 48 x$ .

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Step 1.1.1

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

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x \cdot x^{2} + 14 x^{2} + 48 x}$

Step 1.1.2

Factor $x$ out of $14 x^{2}$ .

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x \cdot x^{2} + x (14 x) + 48 x}$

Step 1.1.3

Factor $x$ out of $48 x$ .

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x \cdot x^{2} + x (14 x) + x \cdot 48}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (14 x)$ .

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x (x^{2} + 14 x) + x \cdot 48}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 14 x) + x \cdot 48$ .

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x (x^{2} + 14 x + 48)}$

Step 1.2

Factor $x^{2} + 14 x + 48$ using the AC method.

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Step 1.2.1

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 $48$ and whose sum is $14$ .

$6, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x ((x + 6) (x + 8))}$

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x^{3} - 64 x$ .

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Step 2.1.1

Cancel the common factor.

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x^{3} - 64 x}{x^{3} - 64 x} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 2.1.2

Rewrite the expression.

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot 1 \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 2.2

Multiply $\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4}$ by $1$ .

$\frac{x^{2} + 10 x + 24}{x^{2} + 5 x + 4} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 3

Factor $x^{2} + 10 x + 24$ 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 $24$ and whose sum is $10$ .

$4, 6$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 6)}{x^{2} + 5 x + 4} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 4

Factor $x^{2} + 5 x + 4$ 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 $4$ and whose sum is $5$ .

$1, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 6)}{(x + 1) (x + 4)} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 6$ .

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Step 5.1.1

Factor $x + 6$ out of $(x + 4) (x + 6)$ .

$\frac{(x + 6) (x + 4)}{(x + 1) (x + 4)} \cdot \frac{x + 1}{x (x + 6) (x + 8)}$

Step 5.1.2

Factor $x + 6$ out of $x (x + 6) (x + 8)$ .

$\frac{(x + 6) (x + 4)}{(x + 1) (x + 4)} \cdot \frac{x + 1}{(x + 6) (x (x + 8))}$

Step 5.1.3

Cancel the common factor.

$\frac{(x + 6) (x + 4)}{(x + 1) (x + 4)} \cdot \frac{x + 1}{(x + 6) (x (x + 8))}$

Step 5.1.4

Rewrite the expression.

$\frac{x + 4}{(x + 1) (x + 4)} \cdot \frac{x + 1}{x (x + 8)}$

Step 5.2

Cancel the common factor of $x + 1$ .

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Step 5.2.1

Cancel the common factor.

$\frac{x + 4}{(x + 1) (x + 4)} \cdot \frac{x + 1}{x (x + 8)}$

Step 5.2.2

Rewrite the expression.

$\frac{x + 4}{x + 4} \cdot \frac{1}{x (x + 8)}$

Step 5.3

Multiply $\frac{x + 4}{x + 4}$ by $\frac{1}{x (x + 8)}$ .

$\frac{x + 4}{(x + 4) (x (x + 8))}$

Step 5.4

Cancel the common factor of $x + 4$ .

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Step 5.4.1

Cancel the common factor.

$\frac{x + 4}{(x + 4) (x (x + 8))}$

Step 5.4.2

Rewrite the expression.

$\frac{1}{x (x + 8)}$