Question

How do you subtract `\frac { 2x ^ { 2} + 7x + 3} { 2x ^ { 2} - 5x - 3} - \frac { ( x - 4) ^ { 3} - 1} { x ^ { 2} - 2x - 15}`?

Answer 1

`((5-x)(x^3-11x^2+28x-48))/((x-3)(x^2-2x-14))`

Explanation:

You must have the same denominator : So you multiply and divide each fraction by the denominator of the other fraction.

`(2x^2+7x+3)/(2x^2-5x-3)-((x-4)^3-1)/(x^2-2x-15)=`

`(2x^2+7x+3)/(2x^2-5x-3)*(x^2-2x-15)/(x^2-2x-15)-`

`((x-4)^3-1)/(x^2-2x-15)*(2x^2-5x-3)/(2x^2-5x-3)=`

`((2x^2+7x+3)(x^2-2x-15)-((x-4)^3-1)(2x^2-5x-3))/((2x^2-5x-3)(x^2-2x-15))`

You do the multiplicantions, expansions, the factoring and get :
(It gets messy so I won't do it here but if you want me to just comment so.)

`-((x-5)cancel((2x+1))(x^3-11x^2+28x-48))/((x-3)cancel((2x+1))(x^2-2x-14))=`

`((5-x)(x^3-11x^2+28x-48))/((x-3)(x^2-2x-14))`