Algebra Examples

$\frac{2 x^{5} + 2 x^{4} - 4 x^{3}}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x^{3}$ out of $2 x^{5} + 2 x^{4} - 4 x^{3}$ .

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Step 1.1.1

Factor $2 x^{3}$ out of $2 x^{5}$ .

$\frac{2 x^{3} (x^{2}) + 2 x^{4} - 4 x^{3}}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1.1.2

Factor $2 x^{3}$ out of $2 x^{4}$ .

$\frac{2 x^{3} (x^{2}) + 2 x^{3} (x) - 4 x^{3}}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1.1.3

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

$\frac{2 x^{3} (x^{2}) + 2 x^{3} (x) + 2 x^{3} (- 2)}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1.1.4

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

$\frac{2 x^{3} (x^{2} + x) + 2 x^{3} (- 2)}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1.1.5

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

$\frac{2 x^{3} (x^{2} + x - 2)}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 1.2

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{2 x^{3} ((x - 1) (x + 2))}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x^{3} - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $6 x$ out of $6 x^{3} - 48 x^{2} + 42 x$ .

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Step 2.1.1

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

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x^{2}) - 48 x^{2} + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2.1.2

Factor $6 x$ out of $- 48 x^{2}$ .

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x^{2}) + 6 x (- 8 x) + 42 x} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2.1.3

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

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x^{2}) + 6 x (- 8 x) + 6 x (7)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2.1.4

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 8 x)$ .

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x^{2} - 8 x) + 6 x (7)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2.1.5

Factor $6 x$ out of $6 x (x^{2} - 8 x) + 6 x (7)$ .

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x^{2} - 8 x + 7)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 2.2

Factor $x^{2} - 8 x + 7$ 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 $7$ and whose sum is $- 8$ .

$- 7, - 1$

Step 2.2.2

Write the factored form using these integers.

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x ((x - 7) (x - 1))} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

$\frac{2 x^{3} (x - 1) (x + 2)}{6 x (x - 7) (x - 1)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Factor $x + 2$ out of $2 x^{3} (x - 1) (x + 2)$ .

$\frac{(x + 2) (2 x^{3} (x - 1))}{6 x (x - 7) (x - 1)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 2) (2 x^{3} (x - 1))}{6 x (x - 7) (x - 1)} \cdot \frac{{(x - 7)}^{2}}{x + 2}$

Step 3.1.3

Rewrite the expression.

$\frac{2 x^{3} (x - 1)}{6 x (x - 7) (x - 1)} \cdot {(x - 7)}^{2}$

Step 3.2

Cancel the common factor of $x - 7$ .

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Step 3.2.1

Factor $x - 7$ out of $6 x (x - 7) (x - 1)$ .

$\frac{2 x^{3} (x - 1)}{(x - 7) (6 x (x - 1))} \cdot {(x - 7)}^{2}$

Step 3.2.2

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

$\frac{2 x^{3} (x - 1)}{(x - 7) (6 x (x - 1))} \cdot ((x - 7) (x - 7))$

Step 3.2.3

Cancel the common factor.

$\frac{2 x^{3} (x - 1)}{(x - 7) (6 x (x - 1))} \cdot ((x - 7) (x - 7))$

Step 3.2.4

Rewrite the expression.

$\frac{2 x^{3} (x - 1)}{6 x (x - 1)} \cdot (x - 7)$

Step 3.3

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

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Step 3.3.1

Factor $2$ out of $2 x^{3} (x - 1)$ .

$\frac{2 (x^{3} (x - 1))}{6 x (x - 1)} \cdot (x - 7)$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

$\frac{2 (x^{3} (x - 1))}{2 (3 x (x - 1))} \cdot (x - 7)$

Step 3.3.2.2

Cancel the common factor.

$\frac{2 (x^{3} (x - 1))}{2 (3 x (x - 1))} \cdot (x - 7)$

Step 3.3.2.3

Rewrite the expression.

$\frac{x^{3} (x - 1)}{3 x (x - 1)} \cdot (x - 7)$

Step 3.4

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

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Step 3.4.1

Factor $x$ out of $x^{3} (x - 1)$ .

$\frac{x (x^{2} (x - 1))}{3 x (x - 1)} \cdot (x - 7)$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $x$ out of $3 x (x - 1)$ .

$\frac{x (x^{2} (x - 1))}{x (3 (x - 1))} \cdot (x - 7)$

Step 3.4.2.2

Cancel the common factor.

$\frac{x (x^{2} (x - 1))}{x (3 (x - 1))} \cdot (x - 7)$

Step 3.4.2.3

Rewrite the expression.

$\frac{x^{2} (x - 1)}{3 (x - 1)} \cdot (x - 7)$

Step 3.5

Cancel the common factor of $x - 1$ .

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Step 3.5.1

Cancel the common factor.

$\frac{x^{2} (x - 1)}{3 (x - 1)} \cdot (x - 7)$

Step 3.5.2

Rewrite the expression.

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

Step 3.6

Apply the distributive property.

$\frac{x^{2}}{3} x + \frac{x^{2}}{3} \cdot - 7$

Step 4

Multiply $\frac{x^{2}}{3} x$ .

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Step 4.1

Combine $\frac{x^{2}}{3}$ and $x$ .

$\frac{x^{2} x}{3} + \frac{x^{2}}{3} \cdot - 7$

Step 4.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.2.1

Multiply $x^{2}$ by $x$ .

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Step 4.2.1.1

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

$\frac{x^{2} x^{1}}{3} + \frac{x^{2}}{3} \cdot - 7$

Step 4.2.1.2

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

$\frac{x^{2 + 1}}{3} + \frac{x^{2}}{3} \cdot - 7$

Step 4.2.2

Add $2$ and $1$ .

$\frac{x^{3}}{3} + \frac{x^{2}}{3} \cdot - 7$

Step 5

Combine $\frac{x^{2}}{3}$ and $- 7$ .

$\frac{x^{3}}{3} + \frac{x^{2} \cdot - 7}{3}$

Step 6

Combine the numerators over the common denominator.

$\frac{x^{3} + x^{2} \cdot - 7}{3}$

Step 7

Reorder factors in $x^{3} + x^{2} \cdot - 7$ .

$\frac{x^{3} - 7 x^{2}}{3}$

Step 8

Factor $x^{2}$ out of $x^{3} - 7 x^{2}$ .

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Step 8.1

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

$\frac{x^{2} x - 7 x^{2}}{3}$

Step 8.2

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

$\frac{x^{2} x + x^{2} \cdot - 7}{3}$

Step 8.3

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

$\frac{x^{2} (x - 7)}{3}$