Chapter 4.6, Problem 33WE

To determine

To find: the factor of the polynomial $4t^2-9y+6$ and verify whether it is prime or not.

Answer to Problem 33WE

  $4t^2-9y+6$ t is a prime polynomial.

Explanation of Solution

Given information:

Given the polynomial $4t^2-9y+6$ .

Calculation:

A non-constant polynomial is prime when it cannot factored into non-constant irreducible polynomials. And thus it cannot be factored with integer, rational and real coefficients.

Appling Sridharacharya formula, $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ for the general two degree polynomial $a{t}^2+bt+c=0$ , to find the roots of the given equation we get,

  $\begin{array}{l}4t^2-9t+6=0\\ \Rightarrow t=\frac{-(-9)\pm \sqrt{{(-9)}^2-(4\times 4\times 6)}}{2\times 4}\\ \Rightarrow t=\frac{9\pm \sqrt{-15}}{8}\\ \Rightarrow t=\frac{9\pm i\sqrt{15}}{8}\end{array}$

Since the roots of the polynomial is imaginary therefore itcannot be factored with integer, rational and real coefficients. Hence it is a prime polynomial.