Simplify ((2s^2-7s-15)/((s+4)^2))÷((s^2-10s+25)/(s+4))

$\frac{2 s^{2} - 7 s - 15}{{(s + 4)}^{2}} \div \frac{s^{2} - 10 s + 25}{s + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 s^{2} - 7 s - 15}{{(s + 4)}^{2}} \cdot \frac{s + 4}{s^{2} - 10 s + 25}$

Step 2

Cancel the common factor of $s + 4$ .

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

$\frac{2 s^{2} - 7 s - 15}{(s + 4) (s + 4)} \cdot \frac{s + 4}{s^{2} - 10 s + 25}$

Cancel the common factor.

$\frac{2 s^{2} - 7 s - 15}{(s + 4) (s + 4)} \cdot \frac{s + 4}{s^{2} - 10 s + 25}$

Rewrite the expression.

$\frac{2 s^{2} - 7 s - 15}{s + 4} \cdot \frac{1}{s^{2} - 10 s + 25}$

Step 3

Multiply $\frac{2 s^{2} - 7 s - 15}{s + 4}$ by $\frac{1}{s^{2} - 10 s + 25}$ .

$\frac{2 s^{2} - 7 s - 15}{(s + 4) (s^{2} - 10 s + 25)}$

Step 4

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 s$ .

$\frac{2 s^{2} - 7 (s) - 15}{(s + 4) (s^{2} - 10 s + 25)}$

Rewrite $- 7$ as $3$ plus $- 10$

$\frac{2 s^{2} + (3 - 10) s - 15}{(s + 4) (s^{2} - 10 s + 25)}$

Apply the distributive property.

$\frac{2 s^{2} + 3 s - 10 s - 15}{(s + 4) (s^{2} - 10 s + 25)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 s^{2} + 3 s) - 10 s - 15}{(s + 4) (s^{2} - 10 s + 25)}$

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

$\frac{s (2 s + 3) - 5 (2 s + 3)}{(s + 4) (s^{2} - 10 s + 25)}$

Factor the polynomial by factoring out the greatest common factor, $2 s + 3$ .

$\frac{(2 s + 3) (s - 5)}{(s + 4) (s^{2} - 10 s + 25)}$

Step 5

Factor using the perfect square rule.

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

$\frac{(2 s + 3) (s - 5)}{(s + 4) (s^{2} - 10 s + 5^{2})}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 s = 2 \cdot s \cdot 5$

Rewrite the polynomial.

$\frac{(2 s + 3) (s - 5)}{(s + 4) (s^{2} - 2 \cdot s \cdot 5 + 5^{2})}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = s$ and $b = 5$ .

$\frac{(2 s + 3) (s - 5)}{(s + 4) {(s - 5)}^{2}}$

Step 6

Cancel the common factor of $s - 5$ and ${(s - 5)}^{2}$ .

Factor $s - 5$ out of $(2 s + 3) (s - 5)$ .

$\frac{(s - 5) (2 s + 3)}{(s + 4) {(s - 5)}^{2}}$

Cancel the common factors.

Factor $s - 5$ out of $(s + 4) {(s - 5)}^{2}$ .

$\frac{(s - 5) (2 s + 3)}{(s - 5) ((s + 4) (s - 5))}$

Cancel the common factor.

$\frac{(s - 5) (2 s + 3)}{(s - 5) ((s + 4) (s - 5))}$

Rewrite the expression.

$\frac{2 s + 3}{(s + 4) (s - 5)}$

Simplify ((2s^2-7s-15)/((s+4)^2))÷((s^2-10s+25)/(s+4))

Related Questions

Support LCMGCF.com by disabling your adblocker.