Video Transcript
Expand 𝑥 minus the square root of
two all cubed.
We have 𝑎 two-toned expression
here. In other words, we have a binomial
which we’re raising to the third power. Since that power is a nonnegative
integer, we know we can apply the binomial theorem. This says that 𝑎 plus 𝑏 to the
𝑛th power where 𝑛 is a nonnegative integer is equal to 𝑎 to the 𝑛th power plus
𝑛 choose one 𝑎 to the power of 𝑛 minus one 𝑏 and so on. And so we begin by comparing our
expression to the general binomial form.
We’re going to let 𝑎 be equal to
𝑥. Then we’re going to be really
careful with 𝑏. A common mistake is to think that
the sign of this term doesn’t matter. In fact, it does. And we’re going to say that since
our general form is 𝑎 plus 𝑏, our value of 𝑏 must be negative root two. And then 𝑛 is equal to three. The first term in our expansion is
𝑎 to the 𝑛th power. So that must be equal to 𝑥
cubed. Then our second term is 𝑛 choose
one, so that’s three choose one, times 𝑎 to the power of 𝑛 minus one, so that’s 𝑥
to the power of three minus one, it’s 𝑥 squared, times negative root two times 𝑏,
which is negative root two.
Our third term is three choose two
times 𝑥 times negative root two squared. And since we know we always have 𝑛
plus one terms in the first line of our expansion, we know that our next term is
going to be the final term. It’s the fourth term. That final term is 𝑏 to the 𝑛th
power, so it’s negative root two cubed. Now three choose one is three
factorial over one factorial times three minus one factorial. And that gives us three. Now three choose two is also three,
so we’re going to replace each of our coefficients three choose one and three choose
two with three.
Once we’ve done that, all that we
need to do is to evaluate our increasing powers of negative root two. Now that’s quite
straightforward. With our second term, it’s just
negative root two. So we can rewrite this as negative
three root two 𝑥 squared. Negative root two squared, though,
is positive two. So our third term becomes three
times two times 𝑥, which is simply six 𝑥. Then our final term can be written
as negative root two times negative root two squared. So that’s negative root two times
two or negative two root two. The expansion then of 𝑥 minus root
two cubed is 𝑥 cubed minus three root two 𝑥 squared plus six 𝑥 minus two root
two.
