Find the product of the coefficients of the terms of the expansion of one minus 𝑥 cubed.
Since we only have one minus 𝑥 cubed, we could expand by multiplying one minus 𝑥 times one minus 𝑥 times one minus 𝑥, then finding all the coefficients and multiplying them together. However, we might recognize that this binomial is almost in the form 𝑎 plus 𝑏 to the 𝑛-power. And when we have something like this and our 𝑛-value is relatively small, we can use Pascal’s triangle for expansion. In Pascal’s triangle, we begin with 𝑛 equals zero and a value of one on top of the triangle. When 𝑛 equals one, we have a row of two ones. When 𝑛 equals two, we have a row one, two, one. And when 𝑛 equals three, we have the row one, three, three, one.
But we should be careful here because this is the case when the binomial is 𝑎 plus 𝑏. If we’re dealing with 𝑎 minus 𝑏, we can still use Pascal’s triangle, but we should know that the signs will be alternating. When 𝑛 equals one, we will have a positive then a negative. When 𝑛 equals two, we have a positive, a negative, a positive. And when 𝑛 equals three, we have a positive, a negative, a positive, and then a negative. And so we have a modified Pascal’s triangle for 𝑎 minus 𝑏 to the 𝑛-power. Our one minus 𝑥 has an 𝑛 of three, which means its coefficient values will be one, negative three, three, and then negative one.
Our goal is to find the product of all of these coefficients, which will be one times negative three times positive three times negative one, which is nine.