Question Video: Expanding a Binomial Using the Binomial Theorem Mathematics

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.