Questions: Use the Rational Zero Theorem to find all the zeros of f(x)=-6x^3+11x^2+50x+8. Enter all the zeros as a comma separated list.

Transcript text: Use the Rational Zero Theorem to find all the zeros of $f(x)=-6 x^{3}+11 x^{2}+50 x+8$. Enter all the zeros as a comma separated list.

Solution

Solution Steps

To find the zeros of the polynomial $ f(x) = -6x^3 + 11x^2 + 50x + 8 $ using the Rational Zero Theorem, we first identify the possible rational zeros. These are the factors of the constant term (8) divided by the factors of the leading coefficient (-6). We then test these possible zeros by substituting them into the polynomial to see which ones yield a result of zero.

Step 1: Identify the Polynomial

We are given the polynomial $ f(x) = -6x^3 + 11x^2 + 50x + 8 $.

Step 2: Apply the Rational Zero Theorem

Using the Rational Zero Theorem, we find the possible rational zeros by taking the factors of the constant term (8) and dividing them by the factors of the leading coefficient (-6). The possible rational zeros are:

\[ \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3} \]

Step 3: Test Possible Zeros

After testing the possible rational zeros, we find that the actual zeros of the polynomial are:

\[ -2, -\frac{1}{6}, 4 \]