Question

How do you simplify `(x ^ 4 - 16y ^ 4)/( ( x² + 4y² ) ( x-2y ))`?

Answer 1

x + 2y

Explanation:

Begin by factorising the numerator, which is a `color(blue)"difference of squares"` and in general factorises as

`color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|))) ........ (A)`

`x^4=(x^2)^2" and " 16y^4=(4y^2)^2`

`rArra=x^2" and " b=4y^2`

Substitute a and b into (A)

`rArrx^4-16y^4=(x^2-4y^2)(x^2+4y^2)`

`rArr((x^2-4y^2)cancel((x^2+4y^2)))/((cancel((x^2+4y^2))(x-2y)`

Now `x^2-4y^2" is also a " color(blue)"difference of squares"`

`x^2=(x)^2" and " 4y^2=(2y)^2rArra=x" and " b=2y`

substitute a and b into (A)

`x^2-4y^2=(x-2y)(x+2y)`

`rArr(cancel((x-2y))(x+2y))/(cancel((x-2y)))=x+2y`