Simplify (x^3-9x)/(x^2+10x+21)*(x^2+7x)/(x^2+5x-24)

$\frac{x^{3} - 9 x}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

Step 1

Simplify the numerator.

Factor $x$ out of $x^{3} - 9 x$ .

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

$\frac{x \cdot x^{2} - 9 x}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

Factor $x$ out of $- 9 x$ .

$\frac{x \cdot x^{2} + x \cdot - 9}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$\frac{x (x^{2} - 9)}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

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

$\frac{x (x^{2} - 3^{2})}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

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 $b = 3$ .

$\frac{x (x + 3) (x - 3)}{x^{2} + 10 x + 21} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

Step 2

Factor $x^{2} + 10 x + 21$ using the AC method.

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 $21$ and whose sum is $10$ .

$3, 7$

Write the factored form using these integers.

$\frac{x (x + 3) (x - 3)}{(x + 3) (x + 7)} \cdot \frac{x^{2} + 7 x}{x^{2} + 5 x - 24}$

Step 3

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

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

$\frac{x (x + 3) (x - 3)}{(x + 3) (x + 7)} \cdot \frac{x \cdot x + 7 x}{x^{2} + 5 x - 24}$

Factor $x$ out of $7 x$ .

$\frac{x (x + 3) (x - 3)}{(x + 3) (x + 7)} \cdot \frac{x \cdot x + x \cdot 7}{x^{2} + 5 x - 24}$

Factor $x$ out of $x \cdot x + x \cdot 7$ .

$\frac{x (x + 3) (x - 3)}{(x + 3) (x + 7)} \cdot \frac{x (x + 7)}{x^{2} + 5 x - 24}$

Step 4

Factor $x^{2} + 5 x - 24$ using the AC method.

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 $5$ .

$- 3, 8$

Write the factored form using these integers.

$\frac{x (x + 3) (x - 3)}{(x + 3) (x + 7)} \cdot \frac{x (x + 7)}{(x - 3) (x + 8)}$

Step 5

Simplify terms.

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $x (x + 3) (x - 3)$ .

$\frac{(x - 3) (x (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x (x + 7)}{(x - 3) (x + 8)}$

Cancel the common factor.

$\frac{(x - 3) (x (x + 3))}{(x + 3) (x + 7)} \cdot \frac{x (x + 7)}{(x - 3) (x + 8)}$

Rewrite the expression.

$\frac{x (x + 3)}{(x + 3) (x + 7)} \cdot \frac{x (x + 7)}{x + 8}$

Cancel the common factor of $x + 7$ .

Factor $x + 7$ out of $(x + 3) (x + 7)$ .

$\frac{x (x + 3)}{(x + 7) (x + 3)} \cdot \frac{x (x + 7)}{x + 8}$

Factor $x + 7$ out of $x (x + 7)$ .

$\frac{x (x + 3)}{(x + 7) (x + 3)} \cdot \frac{(x + 7) x}{x + 8}$

Cancel the common factor.

$\frac{x (x + 3)}{(x + 7) (x + 3)} \cdot \frac{(x + 7) x}{x + 8}$

Rewrite the expression.

$\frac{x (x + 3)}{x + 3} \cdot \frac{x}{x + 8}$

Multiply $\frac{x (x + 3)}{x + 3}$ by $\frac{x}{x + 8}$ .

$\frac{x (x + 3) x}{(x + 3) (x + 8)}$

Step 6

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

$\frac{x^{1} x (x + 3)}{(x + 3) (x + 8)}$

Step 7

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

$\frac{x^{1} x^{1} (x + 3)}{(x + 3) (x + 8)}$

Step 8

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

$\frac{x^{1 + 1} (x + 3)}{(x + 3) (x + 8)}$

Step 9

Add $1$ and $1$ .

$\frac{x^{2} (x + 3)}{(x + 3) (x + 8)}$

Step 10

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{x^{2} (x + 3)}{(x + 3) (x + 8)}$

Rewrite the expression.

$\frac{x^{2}}{x + 8}$

Simplify (x^3-9x)/(x^2+10x+21)*(x^2+7x)/(x^2+5x-24)

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