Video Transcript
Write the coefficients of the terms that result from the expansion of 𝑥 plus 𝑦 to the third power.
We have a few options here. We see that 𝑥 plus 𝑦 cubed is in the form 𝑎 plus 𝑏 to the 𝑛 power. We also notice that we’re only interested in the coefficients of the terms and not all the terms themselves. And the best strategy when the 𝑛-value is small, say less than 10, would be to use Pascal’s triangle to expand this binomial. This is especially true when we’re only interested in the coefficients.
At the top, we have the case when 𝑛 equals zero, there’s only a one. When 𝑛 equals one, we have two ones on either side of this one. When 𝑛 equals two, we have the values one, two, one. Remember that two comes from adding the two coefficients above it; one plus one is two. And then when 𝑛 equals three, the row we’re looking for, we have four coefficients: one, three, three, one. We add one and two to get three for both middle coefficients and then have the ones on either side.
This question is only asking for the coefficients of these terms. And using Pascal’s triangle, the fourth row where 𝑛 equals three, we can say that the coefficients that result from expanding 𝑥 plus 𝑦 cubed will be one, three, three, one.
