Simplify (x^2+7x-8)/(x-7)(x^3-12x^2+35x)/(x^2-3x+2)(x^2-11x+18)/(4x^3+12x^2-160x)

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 11 x + 18}{4 x^{3} + 12 x^{2} - 160 x}$

Step 1

Factor $x^{2} - 11 x + 18$ 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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x^{3} + 12 x^{2} - 160 x}$

Step 2

Simplify the denominator.

Factor $4 x$ out of $4 x^{3} + 12 x^{2} - 160 x$ .

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

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x^{2}) + 12 x^{2} - 160 x}$

Factor $4 x$ out of $12 x^{2}$ .

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x^{2}) + 4 x (3 x) - 160 x}$

Factor $4 x$ out of $- 160 x$ .

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x^{2}) + 4 x (3 x) + 4 x (- 40)}$

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

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x^{2} + 3 x) + 4 x (- 40)}$

Factor $4 x$ out of $4 x (x^{2} + 3 x) + 4 x (- 40)$ .

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x^{2} + 3 x - 40)}$

Factor $x^{2} + 3 x - 40$ 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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Write the factored form using these integers.

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x ((x - 5) (x + 8))}$

$\frac{x^{2} + 7 x - 8}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Step 3

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

$- 1, 8$

Write the factored form using these integers.

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x^{3} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Step 4

Simplify the numerator.

Factor $x$ out of $x^{3} - 12 x^{2} + 35 x$ .

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

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x \cdot x^{2} - 12 x^{2} + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x$ out of $- 12 x^{2}$ .

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x \cdot x^{2} + x (- 12 x) + 35 x}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x$ out of $35 x$ .

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x \cdot x^{2} + x (- 12 x) + x \cdot 35}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x$ out of $x \cdot x^{2} + x (- 12 x)$ .

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x^{2} - 12 x) + x \cdot 35}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x$ out of $x (x^{2} - 12 x) + x \cdot 35$ .

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x^{2} - 12 x + 35)}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x^{2} - 12 x + 35$ 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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Write the factored form using these integers.

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x ((x - 7) (x - 5))}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x - 7) (x - 5)}{x^{2} - 3 x + 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Step 5

Factor $x^{2} - 3 x + 2$ 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 $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x - 7) (x - 5)}{(x - 2) (x - 1)} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Step 6

Simplify terms.

Cancel the common factor of $x - 1$ .

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

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x - 7) (x - 5)}{(x - 1) (x - 2)} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Cancel the common factor.

$\frac{(x - 1) (x + 8)}{x - 7} \cdot \frac{x (x - 7) (x - 5)}{(x - 1) (x - 2)} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Rewrite the expression.

$\frac{x + 8}{x - 7} \cdot \frac{x (x - 7) (x - 5)}{x - 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Cancel the common factor of $x - 7$ .

Factor $x - 7$ out of $x (x - 7) (x - 5)$ .

$\frac{x + 8}{x - 7} \cdot \frac{(x - 7) (x (x - 5))}{x - 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Cancel the common factor.

$\frac{x + 8}{x - 7} \cdot \frac{(x - 7) (x (x - 5))}{x - 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Rewrite the expression.

$(x + 8) \cdot \frac{x (x - 5)}{x - 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $(x + 8) \cdot \frac{x (x - 5)}{x - 2}$ .

$(x + 8) (\frac{x (x - 5)}{x - 2}) \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5) (x + 8)}$

Factor $x + 8$ out of $4 x (x - 5) (x + 8)$ .

$(x + 8) (\frac{x (x - 5)}{x - 2}) \cdot \frac{(x - 9) (x - 2)}{(x + 8) (4 x (x - 5))}$

Cancel the common factor.

$(x + 8) \frac{x (x - 5)}{x - 2} \cdot \frac{(x - 9) (x - 2)}{(x + 8) (4 x (x - 5))}$

Rewrite the expression.

$\frac{x (x - 5)}{x - 2} \cdot \frac{(x - 9) (x - 2)}{4 x (x - 5)}$

Multiply $\frac{x (x - 5)}{x - 2}$ by $\frac{(x - 9) (x - 2)}{4 x (x - 5)}$ .

$\frac{x (x - 5) ((x - 9) (x - 2))}{(x - 2) (4 x (x - 5))}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x - 5) ((x - 9) (x - 2))}{(x - 2) (4 x (x - 5))}$

Rewrite the expression.

$\frac{(x - 5) ((x - 9) (x - 2))}{(x - 2) (4 (x - 5))}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) ((x - 9) (x - 2))}{(x - 2) (4 (x - 5))}$

Rewrite the expression.

$\frac{(x - 9) (x - 2)}{(x - 2) \cdot (4)}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{(x - 9) (x - 2)}{(x - 2) \cdot 4}$

Rewrite the expression.

$\frac{x - 9}{4}$

Simplify (x^2+7x-8)/(x-7)*(x^3-12x^2+35x)/(x^2-3x+2)*(x^2-11x+18)/(4x^3+12x^2-160x)

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Simplify (x^2+7x-8)/(x-7)(x^3-12x^2+35x)/(x^2-3x+2)(x^2-11x+18)/(4x^3+12x^2-160x)