Question

How do you factor `16+ 40x ^ { 2} - 25x ^ { 4}`?

Answer 1

See the explanation below...

Explanation:

These type of polynomials can be factored by substitution.

Let's subtitute `u=x^2` in the above polynomial, so that we get:

`=25u^2+40u+16`

We have to break the `40u` into two terms, let's say `u` and `v` such that their sum must be equal to `40u` and their product must be equal to the product of `25u` and `16`.

Since, `20u+20u=40u` and `20u\cdot 20u = 400u^2`,

we can write:

`=(25u^2+20u)+(20u+16)`

Factor out `5u` from left side terms and `4` from right side terms, so that we get:

`=5u(5u+4)+4(5u+4)`

Factor out common term `(5u+4)`, so that we have left with:

`=(5u+4)(5u+4)`

Substitute back `u=x^2`, and write as a square:

`=(5x^2+4)^2`

Which is our required factored form.