Question

How do you factor completely `4b^4-1024`?

Answer 1

`4(b^2 + 16)(b - 4)(b + 4)`

Explanation:

The first step is to take out the common factor of 4.

`rArr 4(b^4 - 256)`

now `(b^4 - 256)" is a difference of squares "`

since `b^4 = b^2xxb^2" and " 256 = 16xx16`

a difference of squares factors as follows

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

here a `= b^2" and b = 16`

`rArr b^4 - 256 = (b^2 - 16)(b^2 + 16)`

now `b^2 - 16" is also a difference of squares "`

and factors as `(b - 4)(b + 4)`

Putting this together gives

`4b^4 - 1024 = 4(b^2 +16)(b - 4)(b + 4)`