Question

simplify each rational expression. if the rational expression cannot be simplified, so state.

1. $\frac{y^{2}+4y+4}{y^{2}+9y+14}$
2. $\frac{5y-7}{20y+1}$
3. $\frac{x^{2}+2x-15}{x^{2}-2x-35}$
4. $\frac{x^{2}-5x+6}{9-x^{2}}$

Explanation:

Problem 7

Step1: Factor numerator

$y^2+4y+4=(y+2)^2$

Step2: Factor denominator

$y^2+9y+14=(y+2)(y+7)$

Step3: Cancel common factors

$\frac{(y+2)^2}{(y+2)(y+7)}=\frac{y+2}{y+7}$, $y
eq-2,-7$

Problem 8

Step1: Check for common factors

Numerator $5y-7$ and denominator $20y+1$ have no common factors other than 1.

Problem 9

Step1: Factor numerator

$x^2+2x-15=(x+5)(x-3)$

Step2: Factor denominator

$x^2-2x-35=(x+5)(x-7)$

Step3: Cancel common factors

$\frac{(x+5)(x-3)}{(x+5)(x-7)}=\frac{x-3}{x-7}$, $x
eq-5,7$

Problem 10

Step1: Rewrite denominator

$9-x^2=-(x^2-9)=-(x-3)(x+3)$

Step2: Factor numerator

$x^2-5x+6=(x-2)(x-3)$

Step3: Cancel common factors

$\frac{(x-2)(x-3)}{-(x-3)(x+3)}=-\frac{x-2}{x+3}$, $x
eq3,-3$

Answer:

1. $\frac{y+2}{y+7}$, $y

eq-2,-7$

1. Cannot be simplified
2. $\frac{x-3}{x-7}$, $x

eq-5,7$

1. $-\frac{x-2}{x+3}$, $x

eq3,-3$