Use the binomial theorem to find the expansion of 𝑎 plus two 𝑏 to the fourth power.
The binomial theorem is a quick way of evaluating the expansion of some binomial to an exponent. It says that 𝑥 plus 𝑦 to the 𝑛th power is the sum from 𝑘 equals zero to 𝑛 of 𝑛 choose 𝑘 times 𝑥 to the power of 𝑛 minus 𝑘 times 𝑦 to the 𝑘th power. In expanded form, that’s 𝑥 to the 𝑛th power plus 𝑛 choose one times 𝑥 to the power of 𝑛 minus one times 𝑦 plus 𝑛 choose two times 𝑥 to the power of 𝑛 minus two times 𝑦 squared all the way through up to 𝑦 to the 𝑛th power. Let’s compare the general form to our expansion.
We’re going to let 𝑥 be equal to 𝑎, 𝑦 be equal to two 𝑏, and 𝑛 be equal to four. Then, we see that the first term is 𝑎 to the fourth power. Then, our next term is four choose one times 𝑎 to the power of four minus one — well, that’s 𝑎 to the power of three — times two 𝑏. We then have four choose two times 𝑎 to the power of four minus two, so 𝑎 squared, times two 𝑏 squared. We continue this pattern, reducing the power of 𝑎 by one each time and increasing the power of two 𝑏. And so, our fourth term is four choose three times 𝑎 times two 𝑏 cubed. And our final term is two 𝑏 to the fourth power.
Now, let’s evaluate four choose one, four choose two, and four choose three. We say that 𝑛 choose 𝑟 is 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial, which means that four choose one is four factorial over one factorial times four minus one or three factorial. But of course, four factorial is four times three times two times one. One factorial is just one. And three factorial is three times two times one. We see we can simplify by three, two, and one on our numerator and denominator. And so, we’re left with four divided by one, which is simply four. In fact, in a very similar way, we end up with four choose three, also being equal to four.
But what about four choose two? Well, it’s four factorial over two factorial times four minus two, which is two factorial. That’s four times three times two times one over two times one times two times one. Well, we can divide through by four on our numerator and denominator. And we’re left with three times two times one over one times one, which is simply six. And so, we’ve found some part of our coefficient for each term. We’re now going to evaluate the individual exponents of two 𝑏.
So, our very first term is 𝑎 to the fourth power, and our second term becomes four 𝑎 cubed times two 𝑏. Then, we distribute our exponent two over the entire term two 𝑏. And we therefore get six 𝑎 squared times four 𝑏 squared. For our next term, when we distribute the three over two 𝑏, we get two cubed 𝑏 cubed, which is eight 𝑏 cubed. And so, this term is four 𝑎 times eight 𝑏 cubed. Finally, two to the fourth power is 16. So, our last term is 16𝑏 to the fourth power. All that’s left is to simplify each term.
Our first term remains 𝑎 to the fourth power. Then, four times two is eight. So, our second term is eight 𝑎 cubed 𝑏. Six times four is 24. So, our third term is 24𝑎 squared 𝑏 squared. Next, we have 32𝑎𝑏 cubed. And our final term remains as 16𝑏 to the fourth power. And so, we’ve used the binomial theorem to find the expansion of 𝑎 plus two 𝑏 to the fourth power. It’s 𝑎 to the fourth power plus eight 𝑎 cubed 𝑏 plus 24𝑎 squared 𝑏 squared plus 32𝑎𝑏 cubed plus 16𝑏 to the fourth power. Now, it’s important to realize that this only happens for positive integer values of 𝑛.