Questions: The function below has at least one rational zero. Use this fact to find all zeros of the function. g(x)=2x^4-6x^3-19x^2-6x+5 If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

Solution Steps

We start with the polynomial function given by \[ g(x) = 2x^4 - 6x^3 - 19x^2 - 6x + 5. \]

Using the Rational Root Theorem, we identify the possible rational zeros of the polynomial. The possible rational zeros are the factors of the constant term (5) divided by the factors of the leading coefficient (2). This gives us the candidates: \[ \pm 1, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2}. \]

After testing the possible rational zeros, we find that the rational zeros of the polynomial are \[ 5, -1, -\frac{\sqrt{3}}{2} - \frac{1}{2}, -\frac{1}{2} + \frac{\sqrt{3}}{2}. \]

The complete set of zeros for the polynomial \( g(x) \) is \[ 5, -1, -\frac{\sqrt{3}}{2} - \frac{1}{2}, -\frac{1}{2} + \frac{\sqrt{3}}{2}. \]

Final Answer