Question

1. list all of the potential rational roots of $2x^{5}-7x^{4}-5x^{3}+18x^{2}-1$. $2$ $\frac{1}{2}$ $-2$ $1$ $-\frac{1}{2}$ $-1$

Explanation:

To find the potential rational roots of a polynomial \( a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), we use the Rational Root Theorem. The theorem states that any potential rational root \( \frac{p}{q} \) is of the form where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \).

Step 1: Identify the constant term and leading coefficient

For the polynomial \( 2x^5 - 7x^4 - 5x^3 + 18x^2 - 1 \), the constant term \( a_0=- 1 \) and the leading coefficient \( a_n = 2 \).

Step 2: Find the factors of the constant term and leading coefficient

• The factors of the constant term \( -1 \) (denoted as \( p \)) are \( \pm1 \).
• The factors of the leading coefficient \( 2 \) (denoted as \( q \)) are \( \pm1,\pm2 \).

Step 3: Form all possible fractions \( \frac{p}{q} \)

Using the factors of \( p \) and \( q \), we form the possible rational roots:
\[
\frac{\pm1}{\pm1},\frac{\pm1}{\pm2}
\]
Simplifying these fractions, we get:
\[
\pm1,\pm\frac{1}{2}
\]

Answer:

The potential rational roots are \( \frac{1}{2} \), \( 1 \), \( -\frac{1}{2} \), \( - 1 \). So the correct options are:

• \( \boldsymbol{\frac{1}{2}} \)
• \( \boldsymbol{1} \)
• \( \boldsymbol{-\frac{1}{2}} \)
• \( \boldsymbol{-1} \)