Use Pascal’s triangle to determine
the coefficients of the terms that result from the expansion of 𝑥 plus 𝑦 all to
the sixth power.
We’ve been told what method we
should use to find the coefficients, and that’s Pascal’s triangle. And that means our first step will
be to reproduce Pascal’s triangle. Pascal’s triangle helps us to
expand binomials in the form 𝑎 plus 𝑏 all to the 𝑛th power. For the triangle, we start with 𝑛
equals zero. The tip of the triangle is just a
value of one.
Moving on to 𝑛 equals one, we get
the second row that is two values of one. For the third row, we have 𝑛
equals two with the values one, two, one. We get the two in the middle by
adding the ones just above it. Our 𝑛 equals three row will follow
the pattern one, three, three, one. The two three terms come from the
values one plus two above them. When 𝑛 equals four, we have the
row one, four, six, four, one. When 𝑛 equals five, we have the
row one, five, 10, 10, five, one. And for the row 𝑛 equals six, we
have one, six, 15, 20, 15, six, and one.
The binomial we’re expanding is 𝑥
plus 𝑦 all to the sixth power. Our 𝑛-value equals six. And therefore, we’ll have the
coefficients one, six, 15, 20, six, and one.