find real and complex zeros for f(x) x^3+5x^2+17x+13

Question

f(x)=x³+5x²+17x+13  I have to find the real and complex zeros of the following function?

Answer 1

f(-1)=-1+5-17+13=0 so x=-1 is a zero and we can reduce the cubic to a quadratic by synthetically dividing it by the zero:

-1 | 1  5 17     13

      1 -1 -4  | -13

      1  4 13 |    0 = x²+4x+13, which has complex zeroes.

x²+4x=-13,

x²+4x+4=4-13=-9 (completing the square),

(x+2)²=-9, x+2=±3i, x=-2+3i, -2-3i.

SOLUTION

Zeroes are -1, -2+3i, -2-3i.