Chapter 4.6, Problem 28WE

To determine

To determine the factor of polynomial and tell if the polynomial is prime.

Answer to Problem 28WE

The polynomial $6s^2+st-5t^2$ can be factored into $\left(6s-5t\right)\left(s+t\right)$ therefore it is not prime polynomial .

Explanation of Solution

Given:

The polynomial is $6s^2+st-5t^2$ .

Calculation:

A polynomial which cannot be factored into product of two non-constant polynomials is called prime polynomial(irreducible polynomial).

Since, the polynomial is $6s^2+st-5t^2$

Now, splitting $st$ in such a way that product of coefficient of $st$ is $-30$ and there addition is $1$ .

  $\begin{array}{l}6s^2+st-5t^2\\ 6s^2+6st-5st-5t^2\\ here,6-5=1and6\times -5=-30\\ 6s\left(s+t\right)-5t\left(s+t\right)\\ \left(6s-5t\right)\left(s+t\right)\\ \therefore 6s^2+st-5t^2=\left(6s-5t\right)\left(s+t\right)\end{array}$

The polynomial $6s^2+st-5t^2$ can be factored into $\left(6s-5t\right)\left(s+t\right)$ therefore it is not prime polynomial .