Algebra Examples

$\frac{9 x^{3} - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80} \cdot \frac{x^{2} - 16}{x^{2} - 16}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $x^{2} - 16$ .

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Step 1.1.1

Cancel the common factor.

$\frac{9 x^{3} - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80} \cdot \frac{x^{2} - 16}{x^{2} - 16}$

Step 1.1.2

Rewrite the expression.

$\frac{9 x^{3} - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80} \cdot 1$

Step 1.2

Multiply $\frac{9 x^{3} - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$ by $1$ .

$\frac{9 x^{3} - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $9 x$ out of $9 x^{3} - 63 x^{2} - 72 x$ .

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Step 2.1.1

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

$\frac{9 x (x^{2}) - 63 x^{2} - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2.1.2

Factor $9 x$ out of $- 63 x^{2}$ .

$\frac{9 x (x^{2}) + 9 x (- 7 x) - 72 x}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2.1.3

Factor $9 x$ out of $- 72 x$ .

$\frac{9 x (x^{2}) + 9 x (- 7 x) + 9 x (- 8)}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2.1.4

Factor $9 x$ out of $9 x (x^{2}) + 9 x (- 7 x)$ .

$\frac{9 x (x^{2} - 7 x) + 9 x (- 8)}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2.1.5

Factor $9 x$ out of $9 x (x^{2} - 7 x) + 9 x (- 8)$ .

$\frac{9 x (x^{2} - 7 x - 8)}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 2.2

Factor $x^{2} - 7 x - 8$ using the AC method.

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Step 2.2.1

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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 2.2.2

Write the factored form using these integers.

$\frac{9 x ((x - 8) (x + 1))}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

$\frac{9 x (x - 8) (x + 1)}{x + 1} \cdot \frac{x^{2} - 14 x + 40}{x^{2} - 18 x + 80}$

Step 3

Factor $x^{2} - 14 x + 40$ 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 $40$ and whose sum is $- 14$ .

$- 10, - 4$

Step 3.2

Write the factored form using these integers.

$\frac{9 x (x - 8) (x + 1)}{x + 1} \cdot \frac{(x - 10) (x - 4)}{x^{2} - 18 x + 80}$

Step 4

Factor $x^{2} - 18 x + 80$ 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 $80$ and whose sum is $- 18$ .

$- 10, - 8$

Step 4.2

Write the factored form using these integers.

$\frac{9 x (x - 8) (x + 1)}{x + 1} \cdot \frac{(x - 10) (x - 4)}{(x - 10) (x - 8)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 8$ .

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Step 5.1.1

Factor $x - 8$ out of $9 x (x - 8) (x + 1)$ .

$\frac{(x - 8) (9 x (x + 1))}{x + 1} \cdot \frac{(x - 10) (x - 4)}{(x - 10) (x - 8)}$

Step 5.1.2

Factor $x - 8$ out of $(x - 10) (x - 8)$ .

$\frac{(x - 8) (9 x (x + 1))}{x + 1} \cdot \frac{(x - 10) (x - 4)}{(x - 8) (x - 10)}$

Step 5.1.3

Cancel the common factor.

$\frac{(x - 8) (9 x (x + 1))}{x + 1} \cdot \frac{(x - 10) (x - 4)}{(x - 8) (x - 10)}$

Step 5.1.4

Rewrite the expression.

$\frac{9 x (x + 1)}{x + 1} \cdot \frac{(x - 10) (x - 4)}{x - 10}$

Step 5.2

Multiply $\frac{9 x (x + 1)}{x + 1}$ by $\frac{(x - 10) (x - 4)}{x - 10}$ .

$\frac{9 x (x + 1) ((x - 10) (x - 4))}{(x + 1) (x - 10)}$

Step 5.3

Cancel the common factor of $x + 1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{9 x (x + 1) ((x - 10) (x - 4))}{(x + 1) (x - 10)}$

Step 5.3.2

Rewrite the expression.

$\frac{9 x ((x - 10) (x - 4))}{x - 10}$

Step 5.4

Cancel the common factor of $x - 10$ .

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Step 5.4.1

Cancel the common factor.

$\frac{9 x ((x - 10) (x - 4))}{x - 10}$

Step 5.4.2

Divide $9 x (x - 4)$ by $1$ .

$9 x (x - 4)$

Step 5.5

Apply the distributive property.

$9 x \cdot x + 9 x \cdot - 4$

Step 6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

$9 (x \cdot x) + 9 x \cdot - 4$

Step 6.2

Multiply $x$ by $x$ .

$9 x^{2} + 9 x \cdot - 4$

Step 7

Multiply $- 4$ by $9$ .

$9 x^{2} - 36 x$