Chapter 4.6, Problem 35WE

To determine

To find: the factor of the polynomial $12p^2-32pq-5q^2$ and verify whether it is prime or not.

Answer to Problem 35WE

Yes it is a prime polynomial.

Explanation of Solution

Given information:

Given the polynomial $12p^2-32pq-5q^2$ .

Calculation:

A non-constant polynomial is prime when it cannot factored into non-constant irreducible polynomials.

Multiply coefficient of $p^2$ and coefficient of $q^2$ that is $12\times 5=60=2\times 2\times 3\times 5$ . Now express $32$ in terms of $2,2,3,5$ that is, $32=(5\times 3\times 2)+2=30+2$ (note that there is no other choices). Therefore,

  $\begin{array}{l}12p^2-32pq-5q^2\\ =12p^2-(30+2)pq-5q^2\\ =12p^2-30pq-2pq-5q^2\\ =6p(2p-5q)-q(p+5q)\end{array}$

And we cannot split the last expression into multiplication of two non-constant polynomial. Hence given polynomial is irreducible, that is prime polynomial.