Chapter 1.3, Problem 60E

To determine

To calculate: The simplified value of the expression $\left(x+y+z\right)\left(x-y-z\right)$ .

Answer to Problem 60E

The value of the expression $\left(x+y+z\right)\left(x-y-z\right)$ is $x^2-y^2-z^2-2yz$ .

Explanation of Solution

Given information:

The expression $\left(x+y+z\right)\left(x-y-z\right)$ .

Formula used:

To multiply two polynomial expressions, first multiply each term of first polynomial by second polynomial and then add the results. This property is known as distributive property, which is mathematically expressed as,

  $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

Calculation:

Consider the given expression $\left(x+y+z\right)\left(x-y-z\right)$ .

Recall the to multiply two polynomial expressions, first multiply each term of first polynomial by second polynomial and then add the results. This property is known as distributive property, which is mathematically expressed as,

  $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

Apply it,

  $\begin{array}{c}\left(x+y+z\right)\left(x-y-z\right)=x\left(x-y-z\right)+y\left(x-y-z\right)+z\left(x-y-z\right)\\ =x\cdot x+x\cdot \left(-y\right)+x\cdot \left(-z\right)+y\cdot x+y\cdot \left(-y\right)+y\cdot \left(-z\right)+z\cdot x+z\cdot \left(-y\right)+z\cdot \left(-z\right)\\ =x^2-xy-xz+xy-y^2-yz+zx-zy-z^2\end{array}$

Combine the like terms and simplify it further as,

  $\begin{array}{c}\left(x+y+z\right)\left(x-y-z\right)=x^2-xy-xz+xy-y^2-yz+zx-zy-z^2\\ =x^2-y^2-z^2+\left(xy-xy\right)+\left(zx-zx\right)+\left(-yz-zy\right)\\ =x^2-y^2-z^2-2yz\end{array}$

Thus, the value of the expression $\left(x+y+z\right)\left(x-y-z\right)$ is $x^2-y^2-z^2-2yz$ .