Question

Use the rational zeros theorem for all possible rational zeros

Answer 1

The rational zero theorem states that if the polynomial has integer coefficients, then every rational zeros of the function has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Considering the polynomial

$f(x)=2x^3+5x^2+5x+3$

The leading coefficient is the coefficient of the first term, which is 2

The factors are q ±1, ±2

The constant of the polynomial is 3

The factors are p: ±1, ±3

Next, determine all possible values of p/q

First, for p= ±1 and q=±1,±2

$\begin{gathered} \frac{p}{q}=\frac{\pm1}{\pm1}=\pm1 \\ \frac{p}{q}=\frac{\pm1}{\pm2}=\pm\frac{1}{2} \end{gathered}$

Second, for p=±3 and q=±1, ±2

$\begin{gathered} \frac{p}{q}=\frac{\pm3}{\pm1}=\pm3 \\ \frac{p}{q}=\frac{\pm3}{\pm2}=\pm\frac{3}{2} \end{gathered}$

The possible rational zeros of the polynomial are ±1, ±1/2, ±3/2, and ±3