### Video Transcript

Expand π₯ minus the square root of
two all cubed.

We have π two-toned expression
here. In other words, we have a binomial
which weβre raising to the third power. Since that power is a nonnegative
integer, we know we can apply the binomial theorem. This says that π plus π to the
πth power where π is a nonnegative integer is equal to π to the πth power plus
π choose one π to the power of π minus one π and so on. And so we begin by comparing our
expression to the general binomial form.

Weβre going to let π be equal to
π₯. Then weβre going to be really
careful with π. A common mistake is to think that
the sign of this term doesnβt matter. In fact, it does. And weβre going to say that since
our general form is π plus π, our value of π must be negative root two. And then π is equal to three. The first term in our expansion is
π to the πth power. So that must be equal to π₯
cubed. Then our second term is π choose
one, so thatβs three choose one, times π to the power of π minus one, so thatβs π₯
to the power of three minus one, itβs π₯ squared, times negative root two times π,
which is negative root two.

Our third term is three choose two
times π₯ times negative root two squared. And since we know we always have π
plus one terms in the first line of our expansion, we know that our next term is
going to be the final term. Itβs the fourth term. That final term is π to the πth
power, so itβs 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 weβre going to replace each of our coefficients three choose one and three choose
two with three.

Once weβve done that, all that we
need to do is to evaluate our increasing powers of negative root two. Now thatβs quite
straightforward. With our second term, itβs just
negative root two. So we can rewrite this as negative
three root two π₯ squared. Negative root two squared, though,
is positive two. So our third term becomes three
times two times π₯, which is simply six π₯. Then our final term can be written
as negative root two times negative root two squared. So thatβs negative root two times
two or negative two root two. The expansion then of π₯ minus root
two cubed is π₯ cubed minus three root two π₯ squared plus six π₯ minus two root
two.