Use the binomial formula to expand each binomial. (a+2 b)^5...

Name: Problem 19, Chapter 14: Beginning and Intermediate Algebra
Uploaded: 2020-04-09T19:26:35-08:00
Duration: 3 min 41 s
Channel: Julie Silva
Description: Use the binomial formula to expand each binomial. (a+2 b)^5

Key Concepts

Binomial Theorem

The binomial theorem provides a systematic way to expand expressions raised to a power. It states that (a + b)^n can be expanded as the sum of terms in the form C(n, k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem simplifies the process of expansion by directly linking each term's coefficient to the binomial coefficients, avoiding the need for repetitive multiplication.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or “n choose k”, represent the number of ways to choose k elements from a set of n elements without regard to order. They are crucial in the binomial expansion as they determine the weight of each term. These coefficients can be calculated using the formula C(n, k) = n! / (k! * (n-k)!), where '!' denotes factorial.

Exponential Notation in Binomial Expansion

In the context of the binomial theorem, each term in the expansion involves raising the individual components of the binomial to powers that sum to the overall exponent. For instance, in expanding (a + 2b)^5, the variable a is raised to a power that decreases from 5 to 0, while the term (2b) is raised from 0 to 5. Understanding the properties of exponents is essential for correctly computing each term in the expansion.

Transcript

00:01 We're being asked to expand a plus 2b to the fifth power using the binomial theorem.

00:06 So our first term is simply the first term raised to our exponent.

00:11 So it's just going to be a to the fifth power.

00:13 Now, to get the coefficient of the second term, we take our exponent, which is 5, and we divide it by 1 factorial.

00:20 Now, for our variables, a is going to get raised to the 5 minus 1.

00:25 And then our second term, which is 2b, is going to get raised to the first power.

00:30 Now, for our third term, to get its coefficient, we're now going to do 5 times 4 all over 2 factorial.

00:40 Then for our exponents, we're going to have a to the 5 minus 2, and then 2b is going to get raised to the second power.

00:49 So now for our next term, the coefficient, we're going to have 5 times 4 times 3, all over 3 factorial, and then we're going to have a to the 5 minus 3 power, and then 2b raised to the third power.

01:04 And we're going to keep this pattern going until we'd have that last term 2b raised to the 5th.

01:09 Now, as you can see, i ran out of room, so i'm going to make a new line here.

01:12 So for our next term, to get the coefficient, we're going to do 5 times 4 times 3 times 2, all over 4 factorial, and then we're going to have a to the 5 minus 4, and then 2b raised to the 4th power.

01:28 And now, if you notice, our next term, we would have to have 2b raised to the fifth power.

01:33 So our last term will just simply be 2b all raised to the fifth power.

01:39 Well, now we need to simplify this.

01:42 Well, first we have a to the fifth, which is just a to the fifth...

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