Simplify (6x^4-30x^3)/(6x^2-54x+120)*(x^2-5x+4)/(x^3-8x^2+7x)

$\frac{6 x^{4} - 30 x^{3}}{6 x^{2} - 54 x + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Step 1

Factor $6 x^{3}$ out of $6 x^{4} - 30 x^{3}$ .

Factor $6 x^{3}$ out of $6 x^{4}$ .

$\frac{6 x^{3} (x) - 30 x^{3}}{6 x^{2} - 54 x + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $6 x^{3}$ out of $- 30 x^{3}$ .

$\frac{6 x^{3} (x) + 6 x^{3} (- 5)}{6 x^{2} - 54 x + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $6 x^{3}$ out of $6 x^{3} (x) + 6 x^{3} (- 5)$ .

$\frac{6 x^{3} (x - 5)}{6 x^{2} - 54 x + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Step 2

Simplify the denominator.

Factor $6$ out of $6 x^{2} - 54 x + 120$ .

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

$\frac{6 x^{3} (x - 5)}{6 (x^{2}) - 54 x + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $6$ out of $- 54 x$ .

$\frac{6 x^{3} (x - 5)}{6 (x^{2}) + 6 (- 9 x) + 120} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $6$ out of $120$ .

$\frac{6 x^{3} (x - 5)}{6 x^{2} + 6 (- 9 x) + 6 \cdot 20} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

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

$\frac{6 x^{3} (x - 5)}{6 (x^{2} - 9 x) + 6 \cdot 20} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $6$ out of $6 (x^{2} - 9 x) + 6 \cdot 20$ .

$\frac{6 x^{3} (x - 5)}{6 (x^{2} - 9 x + 20)} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Factor $x^{2} - 9 x + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

$\frac{6 x^{3} (x - 5)}{6 ((x - 5) (x - 4))} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{x^{2} - 5 x + 4}{x^{3} - 8 x^{2} + 7 x}$

Step 3

Factor $x^{2} - 5 x + 4$ 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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x^{3} - 8 x^{2} + 7 x}$

Step 4

Simplify the denominator.

Factor $x$ out of $x^{3} - 8 x^{2} + 7 x$ .

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

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x \cdot x^{2} - 8 x^{2} + 7 x}$

Factor $x$ out of $- 8 x^{2}$ .

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x \cdot x^{2} + x (- 8 x) + 7 x}$

Factor $x$ out of $7 x$ .

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x \cdot x^{2} + x (- 8 x) + x \cdot 7}$

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

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x (x^{2} - 8 x) + x \cdot 7}$

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

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x (x^{2} - 8 x + 7)}$

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.

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x ((x - 7) (x - 1))}$

$\frac{6 x^{3} (x - 5)}{6 (x - 5) (x - 4)} \cdot \frac{(x - 4) (x - 1)}{x (x - 7) (x - 1)}$

Step 5

Simplify terms.

Combine.

$\frac{6 x^{3} (x - 5) ((x - 4) (x - 1))}{6 (x - 5) (x - 4) (x (x - 7) (x - 1))}$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{3} (x - 5) ((x - 4) (x - 1))}{6 (x - 5) (x - 4) (x (x - 7) (x - 1))}$

Rewrite the expression.

$\frac{(x^{3} (x - 5)) ((x - 4) (x - 1))}{((x - 5) (x - 4)) (x (x - 7) (x - 1))}$

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

Factor $x$ out of $(x^{3} (x - 5)) ((x - 4) (x - 1))$ .

$\frac{x ((x^{2} (x - 5)) ((x - 4) (x - 1)))}{((x - 5) (x - 4)) (x (x - 7) (x - 1))}$

Cancel the common factors.

Factor $x$ out of $((x - 5) (x - 4)) (x (x - 7) (x - 1))$ .

$\frac{x ((x^{2} (x - 5)) ((x - 4) (x - 1)))}{x (((x - 5) (x - 4)) ((x - 7) (x - 1)))}$

Cancel the common factor.

$\frac{x ((x^{2} (x - 5)) ((x - 4) (x - 1)))}{x (((x - 5) (x - 4)) ((x - 7) (x - 1)))}$

Rewrite the expression.

$\frac{(x^{2} (x - 5)) ((x - 4) (x - 1))}{((x - 5) (x - 4)) ((x - 7) (x - 1))}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x^{2} (x - 5) ((x - 4) (x - 1))}{(x - 5) (x - 4) ((x - 7) (x - 1))}$

Rewrite the expression.

$\frac{(x^{2}) ((x - 4) (x - 1))}{(x - 4) ((x - 7) (x - 1))}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{x^{2} ((x - 4) (x - 1))}{(x - 4) ((x - 7) (x - 1))}$

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (6x^4-30x^3)/(6x^2-54x+120)*(x^2-5x+4)/(x^3-8x^2+7x)

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