How do you find the complex zero of f(x) = x^3 + 13x^2 + 57x + 85

Answer 1

Here are the steps:

1. Start off by finding the real zero, -5.
2. Use synthetic division to find the remaining factor
3. You can find the complex zeros by using the quadratic equation on your result from step 2

Answer 2

You need to factor it. I use the word root instead of the word zero in the explanation below. The terms are synonymous.

First find a rational root (zero) using the rational roots theorem, or by simply looking at a graph. Looking at a graph is the easiest way. You will see that there is a root at -5. To use the rational roots theorem, divide the constant term (85) by the lead coefficient (1). You get 85. The possible factors of 85 are 1,5,17,85 (all plus or minus). If you go and evaluate the polynomial at each of those numbers, you see that -5 is a root.

Since -5 is a root, (x+5) is a factor. Now use long division or synthetic division to divide (x³ +13x² +57x+85) by x+5. You will get x² +8x+17. So the polynomial factors into (x+5)(x²+8x+17). The rational roots can be found by setting the factor x²+8x+17 equal to zero and solving using the quadratic formula. You should get -4+i and -4-i for roots (zeros).