Video Transcript
Subtract six 𝑥 to the fifth power
minus three 𝑦 cubed plus three 𝑧 squared from eight 𝑥 to the fifth power minus
five 𝑦 cubed minus two 𝑧 squared.
To solve this problem, we need to
think carefully about how to write out this subtraction. We’re subtracting six 𝑥 to the
fifth power minus three 𝑦 cubed plus three 𝑧 squared from eight 𝑥 to the fifth
power minus five 𝑦 cubed minus two 𝑧 squared. And this means that our first
expression will be the eight 𝑥 to the fifth power minus five 𝑦 cubed minus two 𝑧
squared. And our second term will be six 𝑥
to the fifth power minus three 𝑦 cubed plus three 𝑧 squared. As we are subtracting the entire
second expression from the first expression, we must put that second expression in
parentheses and then distribute the subtraction across every term in the
expression.
In this step, we pay close
attention to the sign of every term. Six 𝑥 to the fifth power is
positive. And we are going to subtract six 𝑥
to the fifth power. But the three 𝑦 cubed is
negative. And if we want to subtract negative
three 𝑦 cubed, we rewrite that as adding three 𝑦 cubed. The three 𝑧 squared is
positive. We’re subtracting positive three 𝑧
squared, which we can rewrite as minus three 𝑧 squared. And in order to do any subtraction
here, we need to see if we have like terms. Those are terms who have variables
taken to the same exponent.
We have one term, that is, 𝑥 to
the fifth power, and another 𝑥 to the fifth power term. We then have two terms with the
variable 𝑦 cubed and two terms with the variable 𝑧 cubed. If you want, you can regroup so
that the like terms are next to each other in the expression. As you’re doing this, pay close
attention to the signs. And then we remember that, to
combine like terms, we combine their coefficients. Eight 𝑥 to the fifth power minus
six 𝑥 to the fifth power will be eight minus six 𝑥 to the fifth power, which is
two 𝑥 to the fifth power.
For the second terms, we have
negative five 𝑦 cubed plus three 𝑦 cubed. That means we need to combine
negative five and positive three, which is negative two. And for our 𝑧 squared term, we’re
combining the coefficients, negative two and negative three. Negative two minus three is
negative five. Putting all this together, we get
two 𝑥 to the fifth minus two 𝑦 cubed minus five 𝑧 squared.
Before we leave this question,
let’s look at one other way you could solve, starting with our first expression. Instead of writing the second
expression horizontally, you could vertically line the second expression up
underneath the first. Notice that we’ve grouped the 𝑥 to
the fifth terms, the 𝑦 cubed terms, and the 𝑧 squared terms. But again, our biggest challenge
here is to make sure we don’t make any sign mistakes. For the first term, we’re saying
eight 𝑥 to the fifth minus six 𝑥 to the fifth. So we subtract six from eight. And we get two 𝑥 to the fifth.
But our second set of like terms
are not as straightforward. We have negative five 𝑦 cubed. And we are subtracting negative
three 𝑦 cubed, negative five 𝑦 cubed plus three 𝑦 cubed. This is doing that
distribution. Negative five plus three is
negative two. And the variable is 𝑦 cubed. Then we have negative two 𝑧
squared minus positive three 𝑧 squared, which means we’ll have negative two 𝑧
squared minus three 𝑧 squared. Negative two minus three is
negative five. And our variable is 𝑧 squared. And we see here that both methods
will yield the same result.
