Binomial theorem
The binomial theorem is like magic math that helps us to quickly figure out what happens when we multiply a pair of numbers or variables over and over again. It tells us how to find the various terms that show up when we expand a binomial expression, which means a pair of mathematical expressions connected by a plus or minus sign, raised to a power, which means a number that tells us how many times we need to multiply the binomial by itself.
For example, suppose we have the binomial expression (a + b) raised to the second power, which means we need to multiply (a + b) by itself two times. We could do this the long way by writing out every term like, a\*a + a\*b + b\*a + b\*b, then simplifying it down to a\*a + 2\*a\*b + b\*b. However, thanks to the binomial theorem, we can save ourselves a lot of time and effort by using a simple formula that tells us how to find each term based on the coefficients of the two variables, which in this case are both 1.
The binomial theorem formula looks like this: (a + b)\^n = a\^n + n\*a\^(n-1)\*b + (n(n-1)/2!) \* a\^(n-2)\*b\^2 + ... + b\^n. The little caret symbol (^) means "raised to the power of", and the exclamation mark after 2 means "factorial", which is a way of multiplying all the whole numbers between 1 and the given number. For example, 2! = 2\*1 = 2, and 4! = 4\*3\*2\*1 = 24.
So, let's use the binomial theorem formula to find the terms of (a + b)\^2. We start with the first term, which is just a\^2, because there is only one way to choose two a's out of two factors of (a + b). Then we move on to the next term, which is n\*a\^(n-1)\*b. In this case, n is 2, so we have 2\*a\^(2-1)\*b = 2ab. This term represents all the ways to choose one a and one b from two factors of (a + b), which there are two of. Then we have the third term, which is (n(n-1)/2!) \* a\^(n-2)\*b\^2. In this case, n is 2 again, so we have (2(2-1)/2!) \* a\^(2-2)\*b\^2 = b\^2. This term represents all the ways to choose two b's out of two factors of (a + b), which there is only one of.
So, we can simplify our original expression of (a + b)\^2 to a\^2 + 2ab + b\^2. This tells us that when we multiply (a + b) by itself two times, we get three terms, each with a coefficient that corresponds to the number of ways we can choose a's and b's from the factors. The binomial theorem formula can be applied to any binomial expression raised to any power, and it always gives us the correct answer.
For example, suppose we have the binomial expression (a + b) raised to the second power, which means we need to multiply (a + b) by itself two times. We could do this the long way by writing out every term like, a\*a + a\*b + b\*a + b\*b, then simplifying it down to a\*a + 2\*a\*b + b\*b. However, thanks to the binomial theorem, we can save ourselves a lot of time and effort by using a simple formula that tells us how to find each term based on the coefficients of the two variables, which in this case are both 1.
The binomial theorem formula looks like this: (a + b)\^n = a\^n + n\*a\^(n-1)\*b + (n(n-1)/2!) \* a\^(n-2)\*b\^2 + ... + b\^n. The little caret symbol (^) means "raised to the power of", and the exclamation mark after 2 means "factorial", which is a way of multiplying all the whole numbers between 1 and the given number. For example, 2! = 2\*1 = 2, and 4! = 4\*3\*2\*1 = 24.
So, let's use the binomial theorem formula to find the terms of (a + b)\^2. We start with the first term, which is just a\^2, because there is only one way to choose two a's out of two factors of (a + b). Then we move on to the next term, which is n\*a\^(n-1)\*b. In this case, n is 2, so we have 2\*a\^(2-1)\*b = 2ab. This term represents all the ways to choose one a and one b from two factors of (a + b), which there are two of. Then we have the third term, which is (n(n-1)/2!) \* a\^(n-2)\*b\^2. In this case, n is 2 again, so we have (2(2-1)/2!) \* a\^(2-2)\*b\^2 = b\^2. This term represents all the ways to choose two b's out of two factors of (a + b), which there is only one of.
So, we can simplify our original expression of (a + b)\^2 to a\^2 + 2ab + b\^2. This tells us that when we multiply (a + b) by itself two times, we get three terms, each with a coefficient that corresponds to the number of ways we can choose a's and b's from the factors. The binomial theorem formula can be applied to any binomial expression raised to any power, and it always gives us the correct answer.
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