how do you factor x^4+x^3-10x^2+x+1?

Question

x^4+x^3-10x^2+x+1
>_> what are the roots of that.

Answer 1

You try to find a and b to have :

x^4 + x^3 - 10x² + x + 1 = (x² + ax + 1)(x² + bx + 1)
=x^4 + (a + b) x^3 + (ab + 2)x² + (a + b)x +1

then a + b = 1
ab + 2 = -10

b = 1 - a
a(1 - a) + 12 = 0
- a² + a + 12 = 0
a² - a - 12 = 0

a = - 3 or a = 4 then b = 1 - a = 4 or b = - 3 (very good !!)

then the factorization is
(x² - 3x + 1)(x² + 4x + 1)

the roots are
1.5 - 0.5 sqrt(5)
1.5 + 0.5 sqrt(5)
2 - sqrt(3)
2 + sqrt(3)