The problem you're working on asks to find the zeros of the polynomial function:

$f(x) = x^3 - x^2 - 10x + 6$

To solve this, we will first attempt factoring the polynomial by trial or by synthetic division to find possible rational roots.

Step 1: Use the Rational Root Theorem

The Rational Root Theorem tells us that possible rational roots are factors of the constant term (6) divided by factors of the leading coefficient (1, in this case). The possible rational roots are:

$\pm 1, \pm 2, \pm 3, \pm 6$

Step 2: Test Possible Roots

We substitute these values into $f(x)$ to see if any of them are zeros:

• $f(1) = 1^3 - 1^2 - 10(1) + 6 = 1 - 1 - 10 + 6 = -4$ (Not a zero)
• $f(-1) = (-1)^3 - (-1)^2 - 10(-1) + 6 = -1 - 1 + 10 + 6 = 14$ (Not a zero)
• $f(2) = 2^3 - 2^2 - 10(2) + 6 = 8 - 4 - 20 + 6 = -10$ (Not a zero)
• $f(-2) = (-2)^3 - (-2)^2 - 10(-2) + 6 = -8 - 4 + 20 + 6 = 14$ (Not a zero)
• $f(3) = 3^3 - 3^2 - 10(3) + 6 = 27 - 9 - 30 + 6 = -6$ (Not a zero)
• $f(-3) = (-3)^3 - (-3)^2 - 10(-3) + 6 = -27 - 9 + 30 + 6 = 0$

So, $x = -3$ is a zero.

Step 3: Factor the Polynomial

Now that we know $x = -3$ is a zero, we can factor $(x + 3)$ out of the polynomial. To do this, we'll divide $f(x)$ by $x + 3$ using synthetic division.

Synthetic Division:

We divide $f(x) = x^3 - x^2 - 10x + 6$ by $x + 3$:

-3 |  1   -1   -10    6
    |      -3    12   -6
    --------------------
      1   -4     2    0

The quotient is $x^2 - 4x + 2$.

Step 4: Solve the Quadratic

Now, solve $x^2 - 4x + 2 = 0$ using the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(2)}}{2(1)}$

$x = \frac{4 \pm \sqrt{16 - 8}}{2}$

$x = \frac{4 \pm \sqrt{8}}{2}$

$x = \frac{4 \pm 2\sqrt{2}}{2}$

$x = 2 \pm \sqrt{2}$

Step 5: Zeros of the Polynomial

The zeros of the polynomial are:

$x = -3, 2 + \sqrt{2}, 2 - \sqrt{2}$

These are the exact values of the zeros. Enter these values separated by commas into the answer box:

$-3, 2 + \sqrt{2}, 2 - \sqrt{2}$

Would you like further details on any step or have questions?

Related Questions:

1. How does synthetic division work in polynomial division?
2. What is the Rational Root Theorem, and how is it applied?
3. Can all polynomials be factored easily, or do some require approximation techniques?
4. How do you solve quadratic equations using the quadratic formula?
5. What are the different methods to find zeros of polynomials?

Tip: Always check for rational roots using the Rational Root Theorem before attempting more complex methods!