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To find the real zeros of \( f(x) = x^3 - 10x^2 + 28x - 16 \) and use them to factor \( f(x) \), let's proceed as follows:

### Step 1: Find Possible Rational Zeros
We can apply the Rational Root Theorem, which suggests that any rational root of the polynomial will be a factor of the constant term (-16) divided by a factor of the leading coefficient (1). So, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm 16
\]

### Step 2: Test Each Possible Rational Root
We will test these values in \( f(x) \) to see if any are actual roots.

1. **Testing \( x = 2 \):**
   \[
   f(2) = 2^3 - 10 \cdot 2^2 + 28 \cdot 2 - 16 = 8 - 40 + 56 - 16 = 0
   \]
   Since \( f(2) = 0 \), \( x = 2 \) is a root.

### Step 3: Use Synthetic Division
Since \( x = 2 \) is a root, we can divide \( f(x) \) by \( x - 2 \) using synthetic division.

\[
\begin{array}{r|rrr}
2 & 1 & -10 & 28 & -16 \\
   &   & 2  & -16 & 24 \\
\hline
   & 1 & -8 & 12 & 0 \\
\end{array}
\]
The quotient is \( x^2 - 8x + 8 \), so we can rewrite \( f(x) \) as:
\[
f(x) = (x - 2)(x^2 - 8x + 8)
\]

### Step 4: Factor the Quadratic
Now, we need to find the roots of \( x^2 - 8x + 8 = 0 \) using the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 32}}{2} = \frac{8 \pm \sqrt{32}}{2}
\]
\[
x = \frac{8 \pm 4\sqrt{2}}{2} = 4 \pm 2\sqrt{2}
\]
So the real zeros of \( f(x) \) are:
\[
x = 2, \quad x = 4 + 2\sqrt{2}, \quad x = 4 - 2\sqrt{2}
\]

### Step 5: Write \( f(x) \) in Factored Form
Using the real zeros, we can express \( f(x) \) as:
\[
f(x) = (x - 2)\left(x - (4 + 2\sqrt{2})\right)\left(x - (4 - 2\sqrt{2})\right)
\]

### Final Answer
The real zeros are \( x = 2 \), \( x = 4 + 2\sqrt{2} \), and \( x = 4 - 2\sqrt{2} \), and the factored form of \( f(x) \) is:
\[
f(x) = (x - 2)(x - (4 + 2\sqrt{2}))(x - (4 - 2\sqrt{2}))
\]

Would you like more details on any of these steps, or do you have questions?

### Related Questions
1. How can you determine if a polynomial has any complex roots?
2. What is the Rational Root Theorem, and how is it applied?
3. How do you perform synthetic division for a polynomial?
4. What steps can be taken if none of the rational roots work?
5. How do you simplify expressions with square roots in factoring?

**Tip:** When finding zeros of a polynomial, always start with the Rational Root Theorem to narrow down potential solutions.

Find the real zeros of f. Use the real zeros to factor f. f(x) = x^3 - 10x^2 + 28x - 16

Learn how to find the real zeros of the polynomial f(x) = x^3 - 10x^2 + 28x - 16 and use them to factor the expression. This solution includes synthetic division, the Rational Root Theorem, and the quadratic formula, suitable for students in Grades 10-12.

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