Question

use the rational zeros theorem to list all possible rational zeros of the following.
$f(x)=x^{4}-4x^{3}-3x^{2}+5x - 6$
be sure that no value in your list appears more than once.

Explanation:

Step1: Identify leading coefficient and constant term

For the polynomial \( f(x) = x^4 - 4x^3 - 3x^2 + 5x - 6 \), the leading coefficient (coefficient of the highest - degree term) is \( a_n=1 \) (the coefficient of \( x^4 \)) and the constant term is \( a_0 = - 6 \).

Step2: Find factors of constant term and leading coefficient

The factors of the constant term \( a_0=-6 \) are \( \pm1,\pm2,\pm3,\pm6 \). The factors of the leading coefficient \( a_n = 1 \) are \( \pm1 \).

Step3: Apply Rational Zeros Theorem

The Rational Zeros Theorem states that if a polynomial \( f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0 \) has integer coefficients, then every rational zero of \( f(x) \) has the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \).

Since \( q=\pm1 \), the possible rational zeros \( \frac{p}{q} \) are obtained by dividing each factor of \( p \) (the factors of \( a_0 \)) by each factor of \( q \) (the factors of \( a_n \)). So \( \frac{p}{q}=\frac{\pm1}{\pm1},\frac{\pm2}{\pm1},\frac{\pm3}{\pm1},\frac{\pm6}{\pm1} \), which simplifies to \( \pm1,\pm2,\pm3,\pm6 \).

Answer:

\( \pm1,\pm2,\pm3,\pm6 \)