Question

What is the quotient of the rational expression shown below? Make sure your answer is in reduced form. X^2+7X+10/X-2÷X^-25/4X-8

Answer 1

Answer:

Simplified form of $\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} } = \frac{4(X+2)}{(X-5)}$

Step-by-step explanation:

Here, the given expression is

$\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} } = \frac{P(X)}{Q(X)} \\\implies P(X) = {\frac{X^2+7X+10}{X-2} , Q(X) = \frac{X^2-25}{4X-8}$

Now, using the Algebraic Identities:

$(a^2 - b^2) = (a+b)(a-b)$

Simplify P(X) and Q(X) separably:

$P(X) = \frac{X^2+7X+10}{X-2} = \frac{X^2+5X + 2X+10}{X-2} \\\implies P(X) = {\frac{X(X + 5) +2(X+5)}{X-2} \\= P(X) = {\frac{(X+ 5)(X+2)}{X-2} \\\\\implies P(X) = {\frac{(X+ 5)(X+2)}{X-2}$

Similarly, Q(X) = $\frac{X^2-25}{4X-8} = \frac{(X-5)(X+5)}{4X-8} \\\implies Q(X) = \frac{(X-5)(X+5)}{4(X-2)}$

hence, the given fraction is simplified to

$\frac{P(X)}{Q(X)} = \frac{\frac{(X+ 5)(X+2)}{(X-2)} }{\frac{(X-5)(X+5)}{4(X-2)} }$

$\implies \frac{P(X)}{Q(X)} = \frac{\frac{(X+ 5)(X+2)}{(X-2)} }{\frac{(X-5)(X+5)}{4(X-2)} } ={\frac{(X+ 5)(X+2)}{(X-2)} } \times {\frac{ 4(X-2)}{(X-5)(X+5)} = \frac{4(X+2)}{(X-5)}$

Hence, the simplified form of $\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} } = \frac{4(X+2)}{(X-5)}$