Question

According to the Rational Root Theorem, which of the following could be a root of p(x)=3x3−5x2+4? Select all that apply.

Answer 1

SOLUTION

The rational root theorem, also called rational root test theorem state that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, the leading coefficient (the coefficient of the highest power) must be divisible by the denominator of the fraction and the constant term (the one without a variable) must be divisible by the numerator.

Given the polynomial

$p(x)=3x^3-5x^2+4$$\begin{gathered} \text{Leading co}eficient\text{ =3} \\ \text{Constant term =4} \end{gathered}$

The factor of the constant term is

$\pm1,\pm2,\pm4_{}$

The factors of the Leading coefficient are

$\pm1,\pm3$

The root of the p(x) are

$\begin{gathered} \frac{p}{q} \\ \text{where p=factors of the constant term } \\ q=\text{factor of leading coefficient } \end{gathered}$

hence , the possible root of p(x) are

$\pm1,\pm\frac{1}{3},\pm2,\pm\frac{2}{3},\pm4,\pm\frac{4}{3}$

Hence

root of p(x) are 2/3, -2, and 1