Algebra Examples

$\frac{x^{2} + 11 x + 18}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27} \cdot \frac{x^{2} + x - 72}{x^{2} + x - 72}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $x^{2} + x - 72$ .

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Step 1.1.1

Cancel the common factor.

$\frac{x^{2} + 11 x + 18}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27} \cdot \frac{x^{2} + x - 72}{x^{2} + x - 72}$

Step 1.1.2

Rewrite the expression.

$\frac{x^{2} + 11 x + 18}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27} \cdot 1$

Step 1.2

Multiply $\frac{x^{2} + 11 x + 18}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27}$ by $1$ .

$\frac{x^{2} + 11 x + 18}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27}$

Step 2

Factor $x^{2} + 11 x + 18$ using the AC method.

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Step 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 $18$ and whose sum is $11$ .

$2, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{5 - x} \cdot \frac{x^{2} - 2 x - 15}{x^{2} + 12 x + 27}$

Step 3

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{5 - x} \cdot \frac{(x - 5) (x + 3)}{x^{2} + 12 x + 27}$

Step 4

Factor $x^{2} + 12 x + 27$ 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 $27$ and whose sum is $12$ .

$3, 9$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{5 - x} \cdot \frac{(x - 5) (x + 3)}{(x + 3) (x + 9)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 9$ .

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Step 5.1.1

Factor $x + 9$ out of $(x + 2) (x + 9)$ .

$\frac{(x + 9) (x + 2)}{5 - x} \cdot \frac{(x - 5) (x + 3)}{(x + 3) (x + 9)}$

Step 5.1.2

Factor $x + 9$ out of $(x + 3) (x + 9)$ .

$\frac{(x + 9) (x + 2)}{5 - x} \cdot \frac{(x - 5) (x + 3)}{(x + 9) (x + 3)}$

Step 5.1.3

Cancel the common factor.

$\frac{(x + 9) (x + 2)}{5 - x} \cdot \frac{(x - 5) (x + 3)}{(x + 9) (x + 3)}$

Step 5.1.4

Rewrite the expression.

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

Step 5.2

Simplify terms.

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Step 5.2.1

Multiply $\frac{x + 2}{5 - x}$ by $\frac{(x - 5) (x + 3)}{x + 3}$ .

$\frac{(x + 2) ((x - 5) (x + 3))}{(5 - x) (x + 3)}$

Step 5.2.2

Cancel the common factor of $x + 3$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{(x + 2) ((x - 5) (x + 3))}{(5 - x) (x + 3)}$

Step 5.2.2.2

Rewrite the expression.

$\frac{(x + 2) (x - 5)}{5 - x}$

Step 5.2.3

Cancel the common factor of $x - 5$ and $5 - x$ .

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Step 5.2.3.1

Factor $- 1$ out of $x$ .

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

Step 5.2.3.2

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 5.2.3.3

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

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

Step 5.2.3.4

Reorder terms.

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

Step 5.2.3.5

Cancel the common factor.

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

Step 5.2.3.6

Divide $(x + 2) \cdot (- 1)$ by $1$ .

$(x + 2) \cdot (- 1)$

Step 5.2.4

Apply the distributive property.

$x \cdot - 1 + 2 \cdot - 1$

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Move $- 1$ to the left of $x$ .

$- 1 \cdot x + 2 \cdot - 1$

Step 5.2.5.2

Multiply $2$ by $- 1$ .

$- 1 \cdot x - 2$

Step 5.3

Rewrite $- 1 x$ as $- x$ .

$- x - 2$