Question

How do you factor `256m^4-625`?

Answer 1

`=(16m^2+25)(4m+5)(4m-5)`

Explanation:

`256m^4-625`
`=(16m^2)^2-25^2`
`=(16m^2+25)(16m^2-25)`
`=(16m^2+25)((4m)^2-5^2)`
`=(16m^2+25)(4m+5)(4m-5)`

Answer 2

`(4m-5)(4m+5)(16m^2+25)`

Explanation:

This expression is a `color(blue)"difference of squares"` and , in general, factorises as follows.

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

now `256m^4=(16m^2)^2" and " 625=(25)^2`

here a= `16m^2" and " b=25`

substitute these values for a and b into (A)

`rArr256m^4-625=(16m^2-25)(16m^2+25)........ (B)`

Note that the factor `(16m^2-25)" is also "color(blue)"a difference of squares"`

and `16m^2=(4m)^2" while " 25=(5)^2`

here a = 4m and b = 5

`rArr16m^2-25=(4m-5)(4m+5)`

substitute back into (B)

`rArr256m^4-625=(4m-5)(4m+5)(16m^2+25)`