Question

How do you add and simplify `\frac { 9} { 5x ^ { 2} - 2x - 16} + \frac { 6} { 5x ^ { 2} + 28x + 32}`?

Answer 1

`9/(5x^2-2x-16}+6/(5x^2+28x+32}=(15x+24)/((5x^3+18x^2-24x-64)`

Explanation:

As `5x^2-2x-16=5x^2-10x+8x-16=5x(x-2)+4(x-2)=(5x+4)(x-2)`

and `5x^2+28x+32=5x^2+20x+8x+32=5x(x+4)+8(x+4)=(5x+8)(x+4)`

`9/(5x^2-2x-16}+6/(5x^2+28x+32}`

= `(9(5x^2+28x+32}+6((5x^2-2x-16}))/((5x^2-2x-16}(5x^2+28x+32})`

= `(45x^2+252x+288+30x^2-12x-96)/((5x^2-2x-16}(5x^2+28x+32})`

= `(75x^2+240x+192)/((5x^2-2x-16}(5x^2+28x+32})`

= `(3(25x^2+80x+64))/((5x^2-2x-16}(5x^2+28x+32})`

= `(3(25x^2+40x+40x+64))/((5x+4)(x-2)(5x+8)(x+4))`

= `(3(5x(5x+8)+8(5x+8)))/((5x+4)(x-2)(5x+8)(x+4))`

= `(3(5x+8)^2)/((5x+4)(x-2)(5x+8)(x+4))`

= `(3(5x+8))/((5x+4)(x-2)(x+4))`

= `(3(5x+8))/((x+4)(5x^2-2x-16))`

= `(3(5x+8))/((x+4)(5x^2-2x-16))` - see above first line

= `(15x+24)/((5x^3-2x^2-16x+20x^2-8x-64)`

= `(15x+24)/((5x^3+18x^2-24x-64)`