Use the binomial formula to expand each binomial. (5 a-2...

Key Concepts

Binomial Theorem

The binomial theorem provides a formula for expanding expressions that are raised to a power. It states that (x + y)^n can be expanded into a sum of terms in the form C(n, k) * x^(n-k) * y^k, where n is a non-negative integer and k ranges from 0 to n. This theorem simplifies the process of expansion by identifying the coefficients and powers of each term in the expanded form.

Binomial Coefficient

Binomial coefficients, denoted as C(n, k) or 'n choose k', are the numerical factors that multiply the corresponding terms in the binomial expansion. They represent the number of ways to select k elements from a set of n elements without regard to the order in which they are selected. These coefficients not only appear in the expansion of binomials but also play a significant role in combinatorics and probability theory.

Expansion of Binomials

Expanding binomial expressions involves applying the binomial theorem to write the expression in its expanded form. Each term in the expansion is derived from multiplying a binomial coefficient by appropriate powers of each term in the original binomial. This systematic approach allows the transformation of a compact exponentiated expression into a sum of simpler monomials.

Transcript

00:01 So we're being asked to expand the given binomial using the binomial theorem.

00:04 Well, here's how it works.

00:07 The first term, when we do the expansion, will simply be the first term in the binomial, which is 5a, raised to the nth power, which in this case, our exponent is just 4.

00:18 Now, for the second term in this expansion, we first have to figure out the coefficient.

00:22 To do that, we're going to take our n term, which is 4, so we're going to do 4 divided by 1 factorial.

00:30 Then we're going to take our first term in the binomial, which is 5a, and we're going to raise it to the 4 minus 1 power.

00:38 Then we're going to take our second term in the binomial, which is negative 2b, and raise it just to the first power.

00:46 Now, for our third term, to get the coefficient, we're now going to do 4 times 3, because we have to multiply it by 1 less than its original, all over to factorial.

00:57 So notice, we're going to keep this pattern going.

00:59 So in your numerator, you start with your exponent of 4, and then we multiply by 3.

01:04 We go down by 1.

01:05 And in the denominator, your factorial each time goes up by 1.

01:09 Now, we're now going to take the first term in the binomial, 5a, and raise it to the 4 minus 2 power.

01:16 And then we're going to take our second term, which is negative 2b, and raise it to the second power.

01:22 Now, for the next term, we're going to have, for the coefficient, 4 times 3 times 2, all over 3, factorial.

01:32 And then we're going to have 5a raised to the 4 minus 3 power, and then our second term, negative 2b, is going to get raised to the third power.

01:41 Now, what i want you to notice is that last term is now getting raised to the third power, which means in the next term it would be raised to the fourth.

01:49 So whenever we get to that last power, when that last term is going to get raised to that original exponent, we know that's going to be our last term.

01:55 So our last term is simply plus the quantity of negative 2b raised to the fourth power.

02:01 And now we just need to simplify.

02:05 So let's start with that first term.

02:06 We have 5a raised to the fourth power.

02:09 Well, that's just 625a to the 4.

02:14 Next, we have to do 4 divided by 1 factorial, which is just 4...

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