Simplify (2x^5+2x^4-4x^3)/(6x^3-48x^2+42x)*((x-7)^2)/(x+2)

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

Step 1

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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

$- 1, 2$

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.

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

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

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

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

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

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

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

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

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

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

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

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

$- 7, - 1$

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.

Cancel the common factor of $x + 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 7$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Step 4

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

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

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

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

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

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

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

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

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

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 $x^{3} - 7 x^{2}$ .

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

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

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

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

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

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

Simplify (2x^5+2x^4-4x^3)/(6x^3-48x^2+42x)*((x-7)^2)/(x+2)

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