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

1 Answer
Jul 27, 2016

Answer:

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#