Use the binomial theorem to find the expansion of one plus 𝑥 to the fourth power.
We recall that the binomial theorem tells us the expansion of 𝑎 plus 𝑏 to the 𝑛th power for natural numbers 𝑛 is 𝑎 to the 𝑛th power plus 𝑛 choose one times 𝑎 to the power of 𝑛 minus one times 𝑏 plus 𝑛 choose two times 𝑎 to the power of 𝑛 minus two times 𝑏 squared all the way through to 𝑏 to the 𝑛th power. Alternatively, for expressions of the form one plus 𝑥 to the 𝑛th power, where the absolute value of 𝑥 is less than one and 𝑛 is any real number. So essentially that includes negatives and decimals. The expansion is one plus 𝑛𝑥 plus 𝑛 times 𝑛 minus one over one times two 𝑥 squared and so on.
Now, in fact, we’re looking to find the expansion of one plus 𝑥 to the fourth power. That does indeed look like our second example. But actually, since four is a natural number, we can use either form. We’ll use the first formula. We’re going to let 𝑎 be equal to one and 𝑏 be equal to 𝑥. This is to the fourth power. So we let 𝑛 be equal to four. Then the first term in the expansion must be one to the fourth power. The second term is given by four choose one times one cubed times 𝑥. The third term is four choose two times one squared times 𝑥 squared. The fourth term is four choose three times one times 𝑥 cubed. And the fourth and final term is simply 𝑥 to the fourth power.
Let’s look to evaluate four choose one, four choose two, and four choose three. To do this, we recall that 𝑛 choose 𝑟 is given by the formula 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. This means four choose one must be four factorial over one factorial times four minus one factorial. Let’s simplify the denominator to one factorial times three factorial. Then we recall that four factorial is four times three times two times one, which is four times three factorial. So we can rewrite this as four times three factorial over one times three factorial and then divide through by three factorial. And so we see four choose one is equal to four.
Let’s repeat this for four choose two. This time, that’s four factorial over two factorial times four minus two factorial. And so we simplify the denominator to two factorial times two factorial. Next, we write the numerator as four times three times two factorial. And we see that we can divide through by two factorial. Now, two factorial is two. So this simplifies a little further to two times three, which is, of course, simply six. And so four choose two is equal to six.
We’ll repeat this process one more time for four choose three. This time, that’s four factorial over three factorial times one factorial. And if we look really carefully, this is actually the same as four choose one. So it must also be equal to four. Throughout the expansion, one cubed, one squared, and one are simply equal to one. And so we find the binomial expansion of one plus 𝑥 of the fourth power is one plus four 𝑥 plus six 𝑥 squared plus four 𝑥 cubed plus 𝑥 to the fourth power.