Chapter 4.6, Problem 36WE

To determine

To find: the factor of the polynomial $4r^2+16rs-10s^2$ and verify whether it is prime or not.

Answer to Problem 36WE

Yes it is a prime polynomial.

Explanation of Solution

Given information:

Given the polynomial $4r^2+16rs-10s^2$ .

Calculation:

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

We can express given polynomial as,

  $4r^2+16rs-10s^2=2(2r^2+8rs-5s^2)$

Now multiply coefficient of $r^2$ and coefficient of $s^2$ that is $2\times 5=10$ . Now we cannot express the coefficient of $rs$ that is $8$ in terms of $2,5$ . That implies the polynomial $2r^2+8rs-5s^2$ is prime, that is given polynomial is also a prime polynomial.