Simplify (x+5)/(x^3-25x)*(2x^3-11x^2+5x)

$\frac{(x + 5)}{(x^{3} - 25 x)} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Step 1

Simplify the denominator.

Factor $x$ out of $x^{3} - 25 x$ .

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

$\frac{x + 5}{x \cdot x^{2} - 25 x} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Factor $x$ out of $- 25 x$ .

$\frac{x + 5}{x \cdot x^{2} + x \cdot - 25} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 25$ .

$\frac{x + 5}{x (x^{2} - 25)} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Rewrite $25$ as $5^{2}$ .

$\frac{x + 5}{x (x^{2} - 5^{2})} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{x + 5}{x (x + 5) (x - 5)} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Step 2

Simplify terms.

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{x + 5}{x (x + 5) (x - 5)} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Rewrite the expression.

$\frac{1}{x (x - 5)} \cdot (2 x^{3} - 11 x^{2} + 5 x)$

Multiply $\frac{1}{x (x - 5)}$ by $2 x^{3} - 11 x^{2} + 5 x$ .

$\frac{2 x^{3} - 11 x^{2} + 5 x}{x (x - 5)}$

Step 3

Simplify the numerator.

Factor $x$ out of $2 x^{3} - 11 x^{2} + 5 x$ .

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

$\frac{x (2 x^{2}) - 11 x^{2} + 5 x}{x (x - 5)}$

Factor $x$ out of $- 11 x^{2}$ .

$\frac{x (2 x^{2}) + x (- 11 x) + 5 x}{x (x - 5)}$

Factor $x$ out of $5 x$ .

$\frac{x (2 x^{2}) + x (- 11 x) + x \cdot 5}{x (x - 5)}$

Factor $x$ out of $x (2 x^{2}) + x (- 11 x)$ .

$\frac{x (2 x^{2} - 11 x) + x \cdot 5}{x (x - 5)}$

Factor $x$ out of $x (2 x^{2} - 11 x) + x \cdot 5$ .

$\frac{x (2 x^{2} - 11 x + 5)}{x (x - 5)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$\frac{x (2 x^{2} - 11 (x) + 5)}{x (x - 5)}$

Rewrite $- 11$ as $- 1$ plus $- 10$

$\frac{x (2 x^{2} + (- 1 - 10) x + 5)}{x (x - 5)}$

Apply the distributive property.

$\frac{x (2 x^{2} - 1 x - 10 x + 5)}{x (x - 5)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{x ((2 x^{2} - 1 x) - 10 x + 5)}{x (x - 5)}$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

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

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Divide $2 x - 1$ by $1$ .

$2 x - 1$

Simplify (x+5)/(x^3-25x)*(2x^3-11x^2+5x)

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