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. (Enter your answers as a comma-separated list.)
f(x) = 2x4 − 21x3 + 49x2 + 9x − 63

The Rational Zero Test (also known as the Rational Root Theorem) helps in identifying possible rational zeros of a polynomial. According to the theorem, the possible rational zeros are the fractions \( \frac{p}{q} \), where:

- \( p \) is a factor of the constant term (the last term of the polynomial).
- \( q \) is a factor of the leading coefficient (the coefficient of the highest-degree term).

Let's apply the Rational Zero Test to the polynomial:

\[ f(x) = 2x^4 - 21x^3 + 49x^2 + 9x - 63 \]

### Step 1: Identify the leading coefficient and the constant term.
- The leading coefficient is 2 (from the term \( 2x^4 \)).
- The constant term is -63.

### Step 2: List the factors of the constant term and the leading coefficient.
- Factors of -63: \( \pm 1, \pm 3, \pm 7, \pm 9, \pm 21, \pm 63 \)
- Factors of 2: \( \pm 1, \pm 2 \)

### Step 3: Form the possible rational zeros using \( \frac{p}{q} \).
The possible rational zeros are all the combinations of \( \frac{p}{q} \), where \( p \) is a factor of -63 and \( q \) is a factor of 2. The list includes:

\[
\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 7, \pm \frac{7}{2}, \pm 9, \pm \frac{9}{2}, \pm 21, \pm \frac{21}{2}, \pm 63, \pm \frac{63}{2}
\]

### Possible rational zeros:
\[
\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 7, \pm \frac{7}{2}, \pm 9, \pm \frac{9}{2}, \pm 21, \pm \frac{21}{2}, \pm 63, \pm \frac{63}{2}
\]

### Step 4: Verify zeros shown on the graph.
You would check the graph of the polynomial to see which zeros are present. If a graph shows, for example, that \( x = 3 \) and \( x = -1 \) are zeros, you would check if these values are in the list of possible rational zeros (which they are).

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. (Enter your answers as a comma-separated list.) f(x) = 2x^4 − 21x^3 + 49x^2 + 9x − 63

Learn how to apply the Rational Zero Test to the polynomial f(x) = 2x^4 - 21x^3 + 49x^2 + 9x - 63. This guide explains the step-by-step process of listing possible rational zeros and verifying the zeros shown on the graph. Suitable for Grades 10-12.

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