Question

[Lots of Points] What are all of the real roots of the following polynomial?

Answer 1

Notice you can factorize

`x^5+5x^4-5x^3-25x^2+4x+20`

by grouping the terms as

`(x^5-5x^3+4x)+(5x^4-25x^2+20)=x(x^4-5x^2+4)+5(x^4-5x^2+4)`

`\implies f(x)=(x+5)(x^4-5x^2+4)`

Then you know right away that
`x=-5` is a (real) root, so we eliminate C and D.

The remaining quartic can be factored easily:

`x^4-5x^2+4=(x^2)^2-5x^2+4=(x^2-4)(x^2-1)=(x-2)(x+2)(x-1)(x+1)`

which admits four more (also real) roots,
`x=\pm2` and
`x=\pm1`, so the answer is B.