Use the binomial theorem to find the expansion of 𝑎 minus 𝑏 to the fifth power.
𝑎 minus 𝑏 is a binomial expression. It’s an algebraic expression made up of two terms. It’s raised to the fifth power. And we’re told we’re going to need to use the binomial theorem to find its expansion. And so, we recall that the binomial theorem says that for positive integers 𝑛, 𝑥 plus 𝑦 to the 𝑛th power is equal to 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.
Now, this expression can be quite nasty to work with, so we might consider its expanded form. This is 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 to 𝑦 to the 𝑛th power. Notice how the powers of 𝑥 reduce by one each time and the powers of 𝑦 increase.
Let’s compare this formula to our binomial. We see that we can let 𝑥 be equal to 𝑎, 𝑦 is equal to negative 𝑏, and 𝑛, the exponent, is equal to five. This means the first term in our expansion is simply 𝑎 to the fifth power. The second term is 𝑛 choose one, so five choose one, times 𝑎 to the power of five minus one or 𝑎 to the fourth power times negative 𝑏. Remember, the powers of 𝑥, or the powers of 𝑎 here, decrease by one each time, whereas the powers of 𝑦, which is negative 𝑏, increase by one each time. So, our third term is five choose two 𝑎 cubed times negative 𝑏 squared.
Next, we have five choose three 𝑎 squared times negative 𝑏 cubed. Our fifth term is five choose four 𝑎 times negative 𝑏 to the fourth power. And our final term is negative 𝑏 to the fifth power. We’re going to evaluate five choose one, five choose two, five choose three, and five choose four. And so, we recall the formula to help us evaluate 𝑛 choose 𝑟. It’s 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. This means five choose one is five factorial over one factorial times five minus one factorial or five factorial over one factorial times four factorial.
Next, we recall that five factorial is five times four times three times two times one. Similarly, four factorial is four times three times two times one. And we see that we can divide through by four, three, two, and one. In fact, what we’re really doing is dividing through by four factorial. And so, we find five choose one is simply five divided by one, which is five. In much the same way, five choose four also yields a result of five.
Let’s evaluate five choose two. It’s five factorial over two factorial times five minus two factorial. That’s five factorial over two factorial times three factorial. Let’s write five factorial this time instead of as five times four times three times two times one as five times four times three factorial. Then, we see we can divide through by three factorial. By writing two as two times one, we can see we can divide through by two. And five choose two is, therefore, five times two divided by one, which is 10. Five choose three is also 10. And so, we’re ready to find the expansion.
Our first term is still 𝑎 to the fifth power. Our next term is five 𝑎 to the fourth power times negative 𝑏. So, that’s negative five 𝑎 to the fourth power 𝑏. Now, negative 𝑏 squared is positive 𝑏 squared. So, our third term is 10𝑎 cubed 𝑏 squared. When we cube a negative number, we get a negative result. So, our fourth term is negative 10𝑎 squared 𝑏 cubed. We then have five 𝑎𝑏 to the fourth power. And when we raise a negative number to the fifth power, we get a negative result. So, our final term is 𝑏 to the fifth power.
And so, we’re done. We’ve used the binomial theorem to find the expansion of 𝑎 minus 𝑏 to the fifth power. It’s 𝑎 to the fifth power minus five 𝑎 to the fourth power 𝑏 plus 10𝑎 cubed 𝑏 squared minus 10𝑎 squared 𝑏 cubed plus five 𝑎𝑏 to the fourth power minus 𝑏 to the fifth power.