Chapter SH, Problem 5.14EP

To determine

Find the complete factored form of the given polynomial. If it is not then write prime.

Answer to Problem 5.14EP

The factored form of the given polynomial is $\left(x+y\right)\left(x-y\right)\left(2a-3b+c\right)$ .

Explanation of Solution

Given:

The given polynomial is $2a{x}^2-3b{x}^2+c{x}^2-2a{y}^2+3b{y}^2-c{y}^2$ .

Calculation:

Write original expression.

  $2a{x}^2-3b{x}^2+c{x}^2-2a{y}^2+3b{y}^2-c{y}^2$

Group to find a GCF.

  $=2a{x}^2-3b{x}^2+c{x}^2-2a{y}^2+3b{y}^2-c{y}^2$

Factor the GCF.

  $=x^2\left(2a-3b+c\right)-y^2\left(2a-3b+c\right)$

Apply distributive property.

  $=\left(x^2-y^2\right)\left(2a-3b+c\right)$

Expression $\left(x^2-y^2\right)$ is a difference of two squares.

  $a^2-b^2=\left(a+b\right)\left(a-b\right)$

  $=\left(x+y\right)\left(x-y\right)\left(2a-3b+c\right)$

Hence the factored form of the given polynomial is $\left(x+y\right)\left(x-y\right)\left(2a-3b+c\right)$ .