Can you find all zeros (real or complex) of f(x) = x^5-4x^4+4x^3+10x^2-13x-14?

Answer 1

By inspection, -1 is a root of f(x). Divide f(x) synthetically by x+1 to get:

f(x) = (x+1)(x⁴-5x³+9x²+x-14)

-1 is also a root of the second factor. Divide that factor synthetically be x+1 to obtain:

f(x) = (x+1)²(x³-6x²+15x-14)

2 is a root of x³-6x²+15x-14. Dividing synthetically by x-2, we have:

f(x) = (x+1)²(x-2)(x²-4x+7)

Use the quadratic formula to find the remaining roots. They are 2±√3i.