Algebra Examples

$\frac{x^{3} + 15 x^{2} + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} + 15 x^{2} + 54 x$ .

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Step 1.1.1

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

$\frac{x \cdot x^{2} + 15 x^{2} + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1.1.2

Factor $x$ out of $15 x^{2}$ .

$\frac{x \cdot x^{2} + x (15 x) + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1.1.3

Factor $x$ out of $54 x$ .

$\frac{x \cdot x^{2} + x (15 x) + x \cdot 54}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (15 x)$ .

$\frac{x (x^{2} + 15 x) + x \cdot 54}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 15 x) + x \cdot 54$ .

$\frac{x (x^{2} + 15 x + 54)}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 1.2

Factor $x^{2} + 15 x + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Step 1.2.2

Write the factored form using these integers.

$\frac{x ((x + 6) (x + 9))}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

$\frac{x (x + 6) (x + 9)}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 x^{2}$ out of $2 x^{4} + 6 x^{3} - 108 x^{2}$ .

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Step 2.1.1

Factor $2 x^{2}$ out of $2 x^{4}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2.1.2

Factor $2 x^{2}$ out of $6 x^{3}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 2 x^{2} (3 x) - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2.1.3

Factor $2 x^{2}$ out of $- 108 x^{2}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 2 x^{2} (3 x) + 2 x^{2} (- 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2.1.4

Factor $2 x^{2}$ out of $2 x^{2} (x^{2}) + 2 x^{2} (3 x)$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2} + 3 x) + 2 x^{2} (- 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2.1.5

Factor $2 x^{2}$ out of $2 x^{2} (x^{2} + 3 x) + 2 x^{2} (- 54)$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2} + 3 x - 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 2.2

Factor $x^{2} + 3 x - 54$ 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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 2.2.2

Write the factored form using these integers.

$\frac{x (x + 6) (x + 9)}{2 x^{2} ((x - 6) (x + 9))} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $6$ out of $6 x^{2} - 42 x + 36$ .

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Step 3.1.1

Factor $6$ out of $6 x^{2}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2}) - 42 x + 36}{x^{2} + 5 x - 6}$

Step 3.1.2

Factor $6$ out of $- 42 x$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2}) + 6 (- 7 x) + 36}{x^{2} + 5 x - 6}$

Step 3.1.3

Factor $6$ out of $36$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 x^{2} + 6 (- 7 x) + 6 \cdot 6}{x^{2} + 5 x - 6}$

Step 3.1.4

Factor $6$ out of $6 x^{2} + 6 (- 7 x)$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2} - 7 x) + 6 \cdot 6}{x^{2} + 5 x - 6}$

Step 3.1.5

Factor $6$ out of $6 (x^{2} - 7 x) + 6 \cdot 6$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2} - 7 x + 6)}{x^{2} + 5 x - 6}$

Step 3.2

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 3.2.1

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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 3.2.2

Write the factored form using these integers.

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 ((x - 6) (x - 1))}{x^{2} + 5 x - 6}$

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x - 6) (x - 1)}{x^{2} + 5 x - 6}$

Step 4

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 4.2

Write the factored form using these integers.

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x - 6) (x - 1)}{(x - 1) (x + 6)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{x (x + 6) (x + 9) (6 (x - 6) (x - 1))}{2 x^{2} (x - 6) (x + 9) ((x - 1) (x + 6))}$

Step 5.2

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

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Step 5.2.1

Factor $x$ out of $x (x + 6) (x + 9) (6 (x - 6) (x - 1))$ .

$\frac{x (((x + 6) (x + 9)) (6 (x - 6) (x - 1)))}{2 x^{2} (x - 6) (x + 9) ((x - 1) (x + 6))}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $x$ out of $2 x^{2} (x - 6) (x + 9) ((x - 1) (x + 6))$ .

$\frac{x (((x + 6) (x + 9)) (6 (x - 6) (x - 1)))}{x (2 x (x - 6) (x + 9) ((x - 1) (x + 6)))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{x (((x + 6) (x + 9)) (6 (x - 6) (x - 1)))}{x (2 x (x - 6) (x + 9) ((x - 1) (x + 6)))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{((x + 6) (x + 9)) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) ((x - 1) (x + 6))}$

Step 5.3

Cancel the common factor of $x + 6$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(x + 6) (x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) ((x - 1) (x + 6))}$

Step 5.3.2

Rewrite the expression.

$\frac{(x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) (x - 1)}$

Step 5.4

Cancel the common factor of $x + 9$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) (x - 1)}$

Step 5.4.2

Rewrite the expression.

$\frac{6 (x - 6) (x - 1)}{2 x (x - 6) (x - 1)}$

Step 5.5

Cancel the common factor of $6$ and $2$ .

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Step 5.5.1

Factor $2$ out of $6 (x - 6) (x - 1)$ .

$\frac{2 (3 (x - 6) (x - 1))}{2 x (x - 6) (x - 1)}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $2$ out of $2 x (x - 6) (x - 1)$ .

$\frac{2 (3 (x - 6) (x - 1))}{2 ((x (x - 6)) (x - 1))}$

Step 5.5.2.2

Cancel the common factor.

$\frac{2 (3 (x - 6) (x - 1))}{2 ((x (x - 6)) (x - 1))}$

Step 5.5.2.3

Rewrite the expression.

$\frac{3 (x - 6) (x - 1)}{(x (x - 6)) (x - 1)}$

Step 5.6

Cancel the common factor of $x - 6$ .

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Step 5.6.1

Cancel the common factor.

$\frac{3 (x - 6) (x - 1)}{x (x - 6) (x - 1)}$

Step 5.6.2

Rewrite the expression.

$\frac{3 (x - 1)}{(x) (x - 1)}$

Step 5.7

Cancel the common factor of $x - 1$ .

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Step 5.7.1

Cancel the common factor.

$\frac{3 (x - 1)}{x (x - 1)}$

Step 5.7.2

Rewrite the expression.

$\frac{3}{x}$