Chapter 4.6, Problem 27WE

To determine

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

Answer to Problem 27WE

The polynomial $6x^2-7xy-3y^2$ can be factored into $\left(3x+y\right)\left(2x-3y\right)$ therefore it is not prime polynomial .

Explanation of Solution

Given:

The polynomial is $6x^2-7xy-3y^2$ .

Calculation:

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

Since, the polynomial is $6x^2-7xy-3y^2$

Now, splitting $xy$ in such a way that product of coefficient of $xy$ is $-18$ and there addition is $-7$ .

  $\begin{array}{l}6x^2-7xy-3y^2\\ 6x^2-9xy+2xy-3y^2\\ here,-9+2=-7and-9\times 2=-18\\ 3x\left(2x-3y\right)+y\left(2x-3y\right)\\ \left(3x+y\right)\left(2x-3y\right)\\ \therefore 6x^2-7xy-3y^2=\left(3x+y\right)\left(2x-3y\right)\end{array}$

The polynomial $6x^2-7xy-3y^2$ can be factored into $\left(3x+y\right)\left(2x-3y\right)$ therefore it is not prime polynomial .