Question

Find all zeros of the function f(x)= x^3-x^2-4x-2 . Enter the zeros separated by commas. Enter exact values, not decimal approximations.

Answer 1

To find the zeros of the function \( f(x) = x^3 - x^2 - 4x - 2 \), we can use the Rational Root Theorem to test for possible rational roots. The possible rational roots are the factors of the constant term (-2) divided by the factors of the leading coefficient (1). The potential rational roots are \( \pm 1, \pm 2 \).

We will test these values in the function \( f(x) \):

1. **Testing \( x = 1 \)**:
\[
f(1) = (1)^3 - (1)^2 - 4(1) - 2 = 1 - 1 - 4 - 2 = -6 \quad (\text{not a root})
\]

2. **Testing \( x = -1 \)**:
\[
f(-1) = (-1)^3 - (-1)^2 - 4(-1) - 2 = -1 - 1 + 4 - 2 = 0 \quad (\text{is a root})
\]

Since \( x = -1 \) is a root, we will factor \( f(x) \) by using synthetic division with \( x + 1 \):

\[
\begin{array}{r|rrrr}
-1 & 1 & -1 & -4 & -2 \\
& & -1 & 2 & 2 \\
\hline
& 1 & -2 & -2 & 0 \\
\end{array}
\]

After performing synthetic division, we find that:
\[
f(x) = (x + 1)(x^2 - 2x - 2)
\]

Now, we need to find the zeros of the quadratic \( x^2 - 2x - 2 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -2 \), and \( c = -2 \).

Calculating the discriminant:
\[
b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-2) = 4 + 8 = 12
\]

Now substituting into the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{12}}{2 \cdot 1} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}
\]

Thus, the zeros of the function \( f(x) = x^3 - x^2 - 4x - 2 \) are:
\[
x = -1, \quad x = 1 + \sqrt{3}, \quad x = 1 - \sqrt{3}
\]

In conclusion, the exact values of the zeros are:
\[
-1, 1 + \sqrt{3}, 1 - \sqrt{3}
\]