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To solve this problem, we will use the Rational Zero Theorem, which states that any rational root of the polynomial \( f(x) = 4x^5 - 8x^4 - 5x^3 + 10x^2 + x - 2 \) must be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-2) and \( q \) is a factor of the leading coefficient (4).

### Step 1: List factors of the constant term (-2).
The factors of -2 are:
\[
\pm 1, \pm 2
\]

### Step 2: List factors of the leading coefficient (4).
The factors of 4 are:
\[
\pm 1, \pm 2, \pm 4
\]

### Step 3: List all possible rational zeros.
The possible rational zeros are given by \( \frac{p}{q} \), where \( p \) is a factor of -2 and \( q \) is a factor of 4. This gives us the possible rational zeros as:
\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2
\]

### Step 4: Verify which of these potential zeros are actual zeros based on the graph.
Looking at the graph, we can see the polynomial crosses the x-axis at approximately \( x = -1 \) and \( x = 2 \). Therefore, these values are likely zeros of the polynomial.

### Answer:
The rational zeros are:
\[
x = -1, 2
\]

Would you like further details or have any questions?

### Related Questions:
1. How would you determine if there are any imaginary roots for the polynomial?
2. What would be the next step if none of the potential rational zeros worked?
3. How can synthetic division be used to verify these rational zeros?
4. How can the Rational Zero Theorem help in factoring polynomials?
5. What other techniques can be used to find the zeros of a polynomial?

### Tip:
If a polynomial's leading coefficient is 1, the possible rational zeros are simply the factors of the constant term.

Use the Rational Zero Test to list the possible rational zeros of f. Verify that the zeros of f shown on the graph are contained in the list. f(x) = 4x^5 - 8x^4 - 5x^3 + 10x^2 + x - 2.

Learn how to use the Rational Zero Test to list the possible rational zeros of the polynomial f(x) = 4x^5 - 8x^4 - 5x^3 + 10x^2 + x - 2. This step-by-step explanation covers polynomial function analysis and graph interpretation, ideal for Grades 10-12.

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