Question

Perform the indicated operations and simplify.

$\left(x+y+z\right)\left(x-y-z\right)$

Short answer

The solution$\left(x+y+z\right)\left(x-y-z\right)=x^2-y^2-z^2-2yz$

Step-by-step solution

Step 1. Apply the concept of multiplying algebraic expressions (FOIL).

To find the product of polynomials or other algebraic expressions, we need to use the Distributive Property repeatedly. In particular, using it three times on the product of two binomials, we multiply the two factors by multiplying each term in one factor by each term in the other factor and adding these products.

Step 2. Apply the concept of addition and subtraction of polynomials.

We add and subtract polynomials using the properties of real numbers. The idea is to combine like terms (that is, terms with the same variables raised to the same powers) using the Distributive Property.

In subtracting polynomials, we have to remember that if a minus sign precedes an expression in parentheses, then the sign of every term within the parentheses is changed when we remove the parentheses.

Step 3. Analyze the given expression.

Given$\left(x+y+z\right)\left(x-y-z\right)$

Use Distributive property:

$\left(x+y+z\right)\left(x-y-z\right)=x\left(x-y-z\right)+y\left(x-y-z\right)+z\left(x-y-z\right)$

Step 4. Simplify.

Simplify the expression in Step 2:

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