Chapter 4.6, Problem 37WE

To determine

To find: the factor of the polynomial $4x^3+8x^{2}y-5x{y}^2$ and verify whether it is prime or not.

Answer to Problem 37WE

  $4x^3+8x^{2}y-5x{y}^2=x(2x-y)(2x+5y)$

It is a prime polynomial.

Explanation of Solution

Given information:

Given the polynomial $4x^3+8x^{2}y-5x{y}^2$ .

Calculation:

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

Multiply coefficient of $x^3$ and coefficient of $x{y}^2$ that is $4\times 5=20=2\times 2\times 5$ . Now express the coefficient of $x^{2}y$ that is $8$ in terms of $2,2,5$ that is, $8=(5\times 2)-2=10-2$ (note that there is no other choices). Therefore,

  $\begin{array}{l}4x^3+8x^{2}y-5x{y}^2\\ =4x^3+(10-2)x^{2}y-5x{y}^2\\ =4x^3+10x^{2}y-2x^{2}y-5x{y}^2\\ =2x^2(2x+5y)-xy(2x+5y)\\ =(2x^2-xy)(2x+5y)\\ =x(2x-y)(2x+5y)\end{array}$

And we cansplit into non-constant polynomial. Hence given polynomial is reducible, that is not a prime polynomial.