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We are tasked with finding the real zeros of the polynomial function \( f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28 \), using the Rational Zeros Theorem. After identifying the real zeros, we'll factor the polynomial based on those zeros.

### Step 1: Apply the Rational Zeros Theorem
The Rational Zeros Theorem states that any rational zero of a polynomial is of the form:

\[
\frac{p}{q}
\]

where \( p \) is a factor of the constant term (in this case, \( 28 \)), and \( q \) is a factor of the leading coefficient (here, \( 1 \)).

- Factors of \( 28 \): \( \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28 \)
- Factors of \( 1 \): \( \pm 1 \)

Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28 \).

### Step 2: Test potential rational zeros
We can use synthetic division or direct substitution to test these potential zeros.

- **Testing \( x = 2 \):**
  Substitute \( x = 2 \) into the polynomial:
  \[
  f(2) = (2)^4 + 6(2)^3 - 11(2)^2 - 24(2) + 28
  = 16 + 48 - 44 - 48 + 28 = 0
  \]
  So, \( x = 2 \) is a zero.

### Step 3: Polynomial division
Now, divide the polynomial by \( (x - 2) \) using synthetic or long division. After dividing \( f(x) \) by \( (x - 2) \), the quotient is:

\[
x^3 + 8x^2 + 5x - 14
\]

### Step 4: Factor the remaining cubic polynomial
Next, we look for the zeros of \( x^3 + 8x^2 + 5x - 14 \). Let's test more possible rational zeros.

- **Testing \( x = -2 \):**
  Substitute \( x = -2 \):
  \[
  f(-2) = (-2)^3 + 8(-2)^2 + 5(-2) - 14 = -8 + 32 - 10 - 14 = 0
  \]
  So, \( x = -2 \) is another zero.

### Step 5: Factor the quadratic part
Now, divide \( x^3 + 8x^2 + 5x - 14 \) by \( (x + 2) \). The result is:

\[
x^2 + 6x - 7
\]

We can factor \( x^2 + 6x - 7 \) as \( (x - 1)(x + 7) \).

### Step 6: Final factorization
Now we have the complete factorization of \( f(x) \):

\[
f(x) = (x - 2)(x + 2)(x - 1)(x + 7)
\]

### Real Zeros
Thus, the real zeros are:
\[
x = 2, -2, 1, -7
\]

Would you like further explanation on any step or details?

### Relative Questions:
1. How can we confirm that the factored form is correct?
2. What is the process for synthetic division?
3. What if the polynomial had complex zeros? How would we find them?
4. How do you use the Rational Zeros Theorem effectively?
5. Why does polynomial division work to find other factors?

### Tip:
When using the Rational Zeros Theorem, always simplify your list of potential zeros by testing smaller numbers first, as they are easier to calculate.

Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28

Learn how to use the Rational Zeros Theorem to find all real zeros of the polynomial function f(x) = x^4 + 6x^3 - 11x^2 - 24x + 28. This step-by-step solution also demonstrates how to factor the polynomial based on these zeros.

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