Find all zeros of the function. Show all of your work - synthetic division, factoring, quadratic formula, etc. f(x)=3x^3-5x^2+5x-14

Question

The zeros I got from solving this question are: x = -1/4 - √55i/4, -1/4 + √55i/4, and 2. Is this correct?

Answer 1

One of your roots is correct: x = 2. If x =2 is a root, then (x-2) is a factor.

You can use long polynomial division, or synthetic division to find the other polynomial factor.

You should find that f(x) = (x-2)(3x²+x+7).

Now use the quadratic formula to find the roots of g(x) = 3x²+x+7, where a = 3, b=1, c= 7.

The discriminant (value under the radical) will be [1²-4(3)(7)] which indeed is a negative number. so the other two roots are complex imaginary. It looks like that value is -83, so double check your work for finding the two non-real roots.