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The given function is \( f(x) = x^3 - 7x^2 + 8x + 10 \), and we are asked to find all of its zeros.

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem helps us find possible rational roots (factors of the constant term divided by factors of the leading coefficient). Here:
- Constant term \( = 10 \)
- Leading coefficient \( = 1 \)

The possible rational roots are \( \pm 1, \pm 2, \pm 5, \pm 10 \).

### Step 2: Test potential rational roots using synthetic division or direct substitution

Let's test the possible rational roots by substitution into the function:

- For \( x = 1 \):
  \[
  f(1) = 1^3 - 7(1^2) + 8(1) + 10 = 1 - 7 + 8 + 10 = 12 \quad (\text{Not a root})
  \]

- For \( x = -1 \):
  \[
  f(-1) = (-1)^3 - 7(-1)^2 + 8(-1) + 10 = -1 - 7 - 8 + 10 = -6 \quad (\text{Not a root})
  \]

- For \( x = 2 \):
  \[
  f(2) = 2^3 - 7(2^2) + 8(2) + 10 = 8 - 28 + 16 + 10 = 6 \quad (\text{Not a root})
  \]

- For \( x = -2 \):
  \[
  f(-2) = (-2)^3 - 7(-2)^2 + 8(-2) + 10 = -8 - 28 - 16 + 10 = -42 \quad (\text{Not a root})
  \]

- For \( x = 5 \):
  \[
  f(5) = 5^3 - 7(5^2) + 8(5) + 10 = 125 - 175 + 40 + 10 = 0 \quad (\text{This is a root!})
  \]

So, \( x = 5 \) is a root.

### Step 3: Use synthetic division to factor the cubic polynomial

Now that we know \( x = 5 \) is a root, we divide \( f(x) \) by \( (x - 5) \) using synthetic division:

\[
\begin{array}{r|rrrr}
  5 & 1 & -7 & 8 & 10 \\
    &   & 5 & -10 & -10 \\
  \hline
    & 1 & -2 & -2 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 2x - 2 \), and thus we can rewrite the cubic polynomial as:
\[
f(x) = (x - 5)(x^2 - 2x - 2)
\]

### Step 4: Solve the quadratic equation \( x^2 - 2x - 2 = 0 \)

We can solve this using the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)} = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}
\]

Thus, the two other roots are \( x = 1 + \sqrt{3} \) and \( x = 1 - \sqrt{3} \).

### Final Answer:
The zeros of the function are:
\[
x = 5, 1 + \sqrt{3}, 1 - \sqrt{3}
\]

Would you like more details on any step?

Here are 5 related questions to expand on this:
1. How does the Rational Root Theorem work, and how can it be applied to higher-degree polynomials?
2. What is synthetic division, and how is it used to factor polynomials?
3. Can all cubic polynomials be factored into a product of a linear and a quadratic factor?
4. How can you verify that the quadratic roots are exact and not approximations?
5. What other methods exist for finding the roots of a cubic polynomial?

**Tip:** When solving polynomials, always check for possible rational roots first to simplify the factoring process!

Find all zeros of f(x) = x^3 - 7x^2 + 8x + 10. Enter the zeros separated by commas. Enter exact values, not decimal approximations.

Learn how to find all zeros of the cubic polynomial f(x) = x^3 - 7x^2 + 8x + 10. The solution involves using the Rational Root Theorem, synthetic division, and solving a quadratic equation.

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