Use the binomial formula to expand each binomial. (4 a+b)^5...

Name: Problem 23, Chapter 14: Beginning and Intermediate Algebra
Uploaded: 2020-04-09T22:09:53-08:00
Duration: 5 min 3 s
Channel: Julie Silva
Description: Use the binomial formula to expand each binomial. (4 a+b)^5

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that provides a formula for expanding expressions of the form (x + y)^n. It expresses the power as a sum of terms involving binomial coefficients multiplied by appropriate powers of x and y. This systematic approach simplifies expanding polynomial expressions and is essential for understanding more complex topics in algebra and combinatorics.

Binomial Coefficients

Binomial coefficients, often represented as 'n choose k', indicate the number of ways to choose k elements from a set of n elements without regard to order. In the context of the binomial theorem, these coefficients determine the weight or multiplier for each term in the expanded expression. They play a crucial role not only in polynomial expansions but also in probability theory and combinatorics.

Exponents and Powers

Understanding exponents is key when using the binomial theorem. Each term in the expansion involves raising the individual components of the binomial to specific powers—one increasing and the other decreasing as the summation index progresses. Mastery of exponent rules is essential for correctly computing and simplifying each term of the expansion.

Transcript

00:01 So in this problem, we're being asked to expand the binomial 4a plus b raised to the fifth power using the binomial theorem.

00:08 So let's go ahead and work on this.

00:10 So the first term will simply be the first term in that binomial, which is 4a, raise to our exponent, which is fifth.

00:18 So our first term, when we expand this binomial, again, it's always just the first term raised to our exponent.

00:24 Okay.

00:25 So now, to get to the next term, first we have to think about the coefficient.

00:28 Well, we take our exponent, which is 5, and we divide it by 1 factorial.

00:34 Next, we're going to take our first term in the binomial, which is 4a, and we're going to raise it to the 5 minus 1 power.

00:42 Then we're going to take our second term in the binomial, which is b, and raise it simply to the first power.

00:49 Okay, so now we have to get our next term.

00:51 Well, what we're going to do is going to add.

00:54 Now, for our numerator in this next fraction, it's going to be 5 times 4, all over to factorial.

01:02 So if you notice, compared to the first fraction, we multiply 5, and then by 4, we go down by 1.

01:07 But for our factorial, we go up by 1.

01:09 All right, so now what we're going to do is take our first term, which is 4a, and now we're raising it to the 5 minus 2 power.

01:17 And now we're going to take our second term, b, and raise it just to the second power.

01:22 And we're going to continue this pattern until we finally get to that last term, which will have a b to the 5th in it.

01:27 So our next term, we're going to have plus 5 times 4 times 3 all over 3 factorial.

01:36 And then we're going to multiply it by our first term 4a raised to the 5 minus 3 power.

01:43 And then our second term, b, to the third power.

01:47 For our next term, we're going to do 5 times 4 times 3 times 2 all over 4 factorial.

01:55 And that's going to get multiplied by our first term, 4a, raised to the 5 minus 4 power, and then our second term, b, raised to the 4th power.

02:06 Now, if you notice, we're now at b to the 4, so our next term would have b to the 5th.

02:11 So that means we finally got into our last term.

02:13 So our last term will simply be our last term, b, raised to the 5th power.

02:19 Well, now we have to go ahead and simplify this.

02:22 So let's start with our first term, which remember, was 4.

02:25 A raised to the fifth power.

02:28 Well, 4 to the fifth is 1 ,024.

02:31 So our first term is 1 ,024a to the 5th power.

02:36 Now, for our second term, we're going to have plus.

02:39 Well, we have 5 divided by 1 factorial, which is simply 5...

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