Easy approach to the binomial theorem

Let's multiply that times a plus b. Then if you do this, it will be a times a, which is a squared, plus a times b, which is ab, plus b times a, which is another ab, plus b times b, which is b squared.

Easy approach to the binomial theorem

Where there is only one number above, you just carry down the 1. As you might imagine, drawing Pascal's Triangle every time you have to expand a binomial would be a rather long process, especially if the binomial has a large exponent on it. In future videos, we'll do more examples of the binomial theorem and also try to understand why it works. I'll do this. It's 1a to the 4th plus 4a to the 3rd b to the 1st plus 6a squared b squared plus 4ab cubed plus b to the 4th. Now what about a plus b squared? There is an interesting pattern here. There is a symmetry where you have the coefficient, you go 1, 4, 6 for the middle term, and then you go back to 4, and then you go back to 1. Now things are going to get a little bit more interesting. Let's keep going here. Now what is that going to be equal to? That is equal to 4. This is going to be 4 times 3 times 2 times 1 over 2 factorial is 2, over 2 times 2. Let's just review, remind ourselves what n choose k actually means.

Keep going, always adding pairs of numbers from the previous row. Anything that's non-zero to the 0 power, that's just going to be equal to 1.

This is going to be equal to, so we're going to start at k equals 0, so when k equals 0, it's going to be 4 choose 0, 4 choose 0, times a to the 4 minus 0 power, well, that's just going to be a to the 4th power, times b to the 0 power. You have two ab's here, so you could add them together, so it's equal to a squared plus 2ab plus b squared. Now, if you haven't been practicing taking powers of binomials, you might have been tempted to say a squared plus b squared, but that would be incorrect. We were able to figure out what a plus b to the 4th power is. We've expanded it out.

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