Question

Given a polynomial and one of its factors, find the remaining factors of the polynomial. x^4 + 2x^3 – 8x – 16; x + 2

Answer 1

The other factors are:
`(x-2)` and
`(x^2+2x+4)`

Explanation:

Given

Polynomial:
`x^4 + 2x^3 - 8x - 16`

Factor:
`x + 2`

Required

Find other factors

`x^4 + 2x^3 - 8x - 16`

Group into two

`x^4 + 2x^3 - 8x - 16 = (x^4 + 2x^3) - (8x + 16)`

Factorize:

`x^4 + 2x^3 - 8x - 16 = x^3(x + 2) -8 (x + 2)`

Factor out common term

`x^4 + 2x^3 - 8x - 16 = (x^3 -8)(x + 2)`

Rewrite
`x^3- 8` as
`x^3 + 2x^2 - 2x^2+ 4x - 4x - 8`

`x^4 + 2x^3 - 8x - 16 = (x^3 + 2x^2 - 2x^2+ 4x - 4x - 8)(x + 2)`

Rearrange the terms

`x^4 + 2x^3 - 8x - 16 = (x^3 + 2x^2 + 4x - 2x^2 - 4x - 8)(x + 2)`

Factorize

`x^4 + 2x^3 - 8x - 16 = (x(x^2+2x+4)-2(x^2+2x+4))(x + 2)`

Factor out
`(x^2+2x+4)`

`x^4 + 2x^3 - 8x - 16 = (x-2)(x^2+2x+4)(x + 2)`

So, the other factors are:
`(x-2)` and
`(x^2+2x+4)`