Solved Question 9 (0.25 points) The Rational Zeros (or | Chegg.com - Free Printable

Problem Analysis

The Rational Zeros Theorem states that if a polynomial \( P(x) \) has integer coefficients, then any rational zero of the polynomial must be of the form:
\[
\frac{p}{q}
\]
where:
- \( p \) is a factor of the constant term (the term with no \( x \)),
- \( q \) is a factor of the leading coefficient (the coefficient of the highest power of \( x \)).

We need to match each polynomial with its list of possible rational zeros by identifying the factors of the constant term and the leading coefficient.

Step-by-Step Solution

#### 1. Polynomial: \( P(x) = 4x^6 - 24x^4 + 13x^3 + 17x - 5 \)
- Leading coefficient: \( 4 \)
- Factors of \( 4 \): \( \pm 1, \pm 2, \pm 4 \)
- Constant term: \( -5 \)
- Factors of \( -5 \): \( \pm 1, \pm 5 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{5}{1}, \pm \frac{5}{2}, \pm \frac{5}{4}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}
\]
- Match: List 1

#### 2. Polynomial: \( P(x) = 5x^6 - 24x^4 + 13x^3 + 5x^2 - 4 \)
- Leading coefficient: \( 5 \)
- Factors of \( 5 \): \( \pm 1, \pm 5 \)
- Constant term: \( -4 \)
- Factors of \( -4 \): \( \pm 1, \pm 2, \pm 4 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{2}{1}, \pm \frac{2}{5}, \pm \frac{4}{1}, \pm \frac{4}{5}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 4, \pm \frac{4}{5}
\]
- Match: List 6

#### 3. Polynomial: \( P(x) = 6x^5 + 13x^3 + 5x^2 + 17x - 4 \)
- Leading coefficient: \( 6 \)
- Factors of \( 6 \): \( \pm 1, \pm 2, \pm 3, \pm 6 \)
- Constant term: \( -4 \)
- Factors of \( -4 \): \( \pm 1, \pm 2, \pm 4 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{2}{1}, \pm \frac{2}{2}, \pm \frac{2}{3}, \pm \frac{2}{6}, \pm \frac{4}{1}, \pm \frac{4}{2}, \pm \frac{4}{3}, \pm \frac{4}{6}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{2}{3}
\]
- Match: List 3

#### 4. Polynomial: \( P(x) = 5x^4 - 24x^3 + 3x^2 + 17x - 6 \)
- Leading coefficient: \( 5 \)
- Factors of \( 5 \): \( \pm 1, \pm 5 \)
- Constant term: \( -6 \)
- Factors of \( -6 \): \( \pm 1, \pm 2, \pm 3, \pm 6 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{2}{1}, \pm \frac{2}{5}, \pm \frac{3}{1}, \pm \frac{3}{5}, \pm \frac{6}{1}, \pm \frac{6}{5}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 3, \pm \frac{3}{5}, \pm 6, \pm \frac{6}{5}
\]
- Match: List 5

#### 5. Polynomial: \( P(x) = 4x^5 - 24x^4 + 5x^2 + 17x - 6 \)
- Leading coefficient: \( 4 \)
- Factors of \( 4 \): \( \pm 1, \pm 2, \pm 4 \)
- Constant term: \( -6 \)
- Factors of \( -6 \): \( \pm 1, \pm 2, \pm 3, \pm 6 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{2}{1}, \pm \frac{2}{2}, \pm \frac{2}{4}, \pm \frac{3}{1}, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{6}{1}, \pm \frac{6}{2}, \pm \frac{6}{4}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 3, \pm 6, \pm \frac{3}{2}, \pm \frac{3}{4}
\]
- Match: List 4

#### 6. Polynomial: \( P(x) = 6x^4 - 24x^3 + 13x^2 + 17x - 5 \)
- Leading coefficient: \( 6 \)
- Factors of \( 6 \): \( \pm 1, \pm 2, \pm 3, \pm 6 \)
- Constant term: \( -5 \)
- Factors of \( -5 \): \( \pm 1, \pm 5 \)
- Possible rational zeros:
\[
\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{5}{1}, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}
\]
Simplifying, we get:
\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}
\]
- Match: List 2

Final Answer

\[
\boxed{
\begin{aligned}
1 & : P(x) = 4x^6 - 24x^4 + 13x^3 + 17x - 5 \\
2 & : P(x) = 6x^4 - 24x^3 + 13x^2 + 17x - 5 \\
3 & : P(x) = 6x^5 + 13x^3 + 5x^2 + 17x - 4 \\
4 & : P(x) = 4x^5 - 24x^4 + 5x^2 + 17x - 6 \\
5 & : P(x) = 5x^4 - 24x^3 + 3x^2 + 17x - 6 \\
6 & : P(x) = 5x^6 - 24x^4 + 13x^3 + 5x^2 - 4 \\
\end{aligned}
}
\]