Help with finding polynomial equation?

x^4+3x^3-5x^2-15x=0

Since there is no constant term we can immediately say

x(x^3+3x^2-5x-15)=0

so x = 0 is a root.

That leaves

(x^3+3x^2-5x-15) = 0

Now we can factorise using the BOX METHOD

(GOOGLE this really simple neat method, can also be used in EXCEL)

We are aiming for something like

(x+a)(Some quadratic) = 0

As it turns out this is simply:

(x+3)(x^2-5) = 0

so x = -3 is the next root

and we are left with

(X^2-5) = 0

so x = ±√5 are the remaining two roots.

ALL roots are real.

You may factor by grouping:

x^4 + 3x^3 - 5x^2 - 15x = x^3(x+3) - 5x(x + 3) = (x^3 - 5x)(x + 3) = x(x^2 - 5)(x + 3) = 0

and so:

x = 0, ±√(5), -3