### Video Transcript

Determine which of the following
expressions is equivalent to π₯ plus π¦ plus π§ squared. Is it (A) two π₯ squared plus two
π¦ squared plus two π§ squared? Is it (B) two π₯ squared plus two
π¦ squared plus two π§ squared plus π₯π¦ plus π₯π§ plus π¦π§? Is it (C) π₯ squared plus π¦
squared plus π§ squared plus π₯π¦ plus π₯π§ plus π¦π§? (D) π₯ squared plus π¦ squared plus
π§ squared. Or is it (E) π₯ squared plus π¦
squared plus π§ squared plus two π₯π¦ plus two π₯π§ plus two π¦π§?

In order to decide which of the
expressions is equivalent to π₯ plus π¦ plus π§ squared, weβre going to distribute
the parentheses or expand the brackets. Now, in doing so, we need to be
really careful. When we square an expression, we
multiply it by itself. And so π₯ plus π¦ plus π§ squared
is π₯ plus π¦ plus π§ times π₯ plus π¦ plus π§. A common mistake here is to simply
square each of the individual terms. That would give us answer (D),
which is in fact incorrect. So how are we going to distribute
these parentheses? Well, weβll need to be quite
methodical.

Letβs begin by taking the π₯ in the
first set of parentheses and multiplying it by everything in the second. We get π₯ times π₯, which is π₯
squared; π₯ times π¦, which is π₯π¦; then π₯ times π§, which is π₯π§. Weβll now repeat this process with
the π¦. π¦ times π₯ is π₯π¦, π¦ times π¦ is
π¦ squared, and π¦ times π§ is π¦π§. Thereβs just one term left; itβs
this π§. π§ times π₯ is π₯π§. We then multiply π§ by π¦ to get
π¦π§. And finally, π§ times π§ is π§
squared.

So letβs simplify a little
further. We have π₯ squared, π¦ squared, and
π§ squared. Then π₯π¦ plus π₯π¦ is two π₯π¦,
π₯π§ plus π₯π§ is two π₯π§, and finally, π¦π§ plus π¦π§ is two π¦π§. And so when we distribute π₯ plus
π¦ plus π§ squared, we get π₯ squared plus π¦ squared plus π§ squared plus two π₯π¦
plus two π₯π§ plus two π¦π§. And so we see that the expression
that is equivalent to π₯ plus π¦ plus π§ squared is (E).

In fact, what we have is an
identity. We can say that for all values of
π₯, π¦, and π§, π₯ plus π¦ plus π§ squared is equal to π₯ squared plus π¦ squared
plus π§ squared plus two π₯π¦ plus two π₯π§ plus two π¦π§.