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

Simplified form of
`((X^2+7X+10)/(X-2) )/((X^2-25)/(X-8) )   = (4(X+2))/((X-5))`

Explanation:

Here, the given expression is

`((X^2+7X+10)/(X-2) )/((X^2-25)/(X-8) )   = (P(X))/(Q(X)) \\\implies P(X) = {(X^2+7X+10)/(X-2) , Q(X) = (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) = (X^2+7X+10)/(X-2) =   (X^2+5X + 2X+10)/(X-2) \\\implies P(X) = {(X(X + 5) +2(X+5))/(X-2)   \\= P(X) = {((X+ 5)(X+2))/(X-2) \\\\\implies  P(X) =  {((X+ 5)(X+2))/(X-2)`

Similarly, Q(X) =
`(X^2-25)/(4X-8)  = ((X-5)(X+5))/(4X-8) \\\implies Q(X) =  ((X-5)(X+5))/(4(X-2))`

hence, the given fraction is simplified to

`(P(X))/(Q(X))  = (((X+ 5)(X+2))/((X-2)) )/(((X-5)(X+5))/(4(X-2)) )`

`\implies (P(X))/(Q(X))  = (((X+ 5)(X+2))/((X-2)) )/(((X-5)(X+5))/(4(X-2)) )  ={((X+ 5)(X+2))/((X-2)) } *  {( 4(X-2))/((X-5)(X+5))   = (4(X+2))/((X-5))`

Hence, the simplified form of
`((X^2+7X+10)/(X-2) )/((X^2-25)/(X-8) )   = (4(X+2))/((X-5))`