Chapter 4.5, Problem 41WE

To determine

To factor: the provided polynomial

Answer to Problem 41WE

The factor of the provided polynomial is $\left(u-v\right)\left(u+v-2\right)$

Explanation of Solution

Given information: the polynomial provided in the question,

  $u^2-v^2-2u-2v$

Calculation:

Use thedifference of squares on first and second terms and third and fourth terms have 2 in common,

  $\begin{array}{l}{u}^2-v^2-2u-2v\\ =\left(u-v\right)\left(u+v\right)-2u-2v\\ =\left(u-v\right)\left(u+v\right)-2\left(u-v\right)\end{array}$

  $\left(u-v\right)$ is common in first and second term,

  $\begin{array}{l}\left(u-v\right)\left(u+v\right)-2\left(u-v\right)\\ =\left(u-v\right)\left(u+v-2\right)\end{array}$

Hence, the factor of the provided polynomial is $\left(u-v\right)\left(u+v-2\right)$