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The question is asking for the real zeros of the polynomial function \( f(x) = 5x^4 + 4x^3 - 11x^2 - 8x + 2 \) using the Rational Zeros Theorem.

### Rational Zeros Theorem:
The Rational Zeros Theorem helps to identify possible rational zeros of a polynomial. It states that if a polynomial has a rational zero \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient, then all possible rational zeros can be tested.

For the given polynomial:
- The constant term is \( 2 \), so the factors of \( 2 \) are \( \pm 1, \pm 2 \).
- The leading coefficient is \( 5 \), so the factors of \( 5 \) are \( \pm 1, \pm 5 \).

Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm \frac{1}{5}, \pm \frac{2}{5} \).

### Next Step: Testing Possible Zeros
We substitute each possible rational zero into the polynomial to check whether it is a zero.

1. **\( x = 1 \):**
   \[
   f(1) = 5(1)^4 + 4(1)^3 - 11(1)^2 - 8(1) + 2 = 5 + 4 - 11 - 8 + 2 = -8 \quad (\text{not a zero})
   \]

2. **\( x = -1 \):**
   \[
   f(-1) = 5(-1)^4 + 4(-1)^3 - 11(-1)^2 - 8(-1) + 2 = 5 - 4 - 11 + 8 + 2 = 0 \quad (\text{zero found})
   \]

Thus, \( x = -1 \) is a zero.

### Synthetic Division:
Now, we use synthetic division to divide the polynomial \( f(x) \) by \( (x + 1) \):

\[
\text{Dividing } 5x^4 + 4x^3 - 11x^2 - 8x + 2 \text{ by } (x + 1):
\]

After performing the synthetic division, the quotient is:

\[
5x^3 - x^2 - 12x + 2
\]

Now, we repeat the Rational Zeros Theorem for this quotient to find more zeros.

Would you like me to continue solving the problem or clarify anything? 

### Follow-Up Questions:
1. What is the next step after finding the first zero?
2. How does synthetic division work with polynomials?
3. Can a polynomial have complex or imaginary zeros?
4. How do we identify irrational zeros in a polynomial?
5. What happens if a polynomial has no real zeros?

### Tip:
When using the Rational Zeros Theorem, always test smaller factors first, as they are easier to work with manually.

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) = 5x^4 + 4x^3 - 11x^2 - 8x + 2

Learn how to find the real zeros of the polynomial function f(x) = 5x^4 + 4x^3 - 11x^2 - 8x + 2 using the Rational Zeros Theorem. Includes step-by-step synthetic division and testing of possible rational zeros.

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