Expand each expression using the Binomial Theorem. (2 x+3)^5...

Name: Problem 22, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-05-13T23:49:52-08:00
Duration: 4 min 46 s
Channel: John Vanschoick
Description: Expand each expression using the Binomial Theorem. (2 x+3)^5

Key Concepts

Binomial Theorem

The Binomial Theorem provides a systematic method for expanding expressions in the form (a + b)^n. It states that this expansion can be written as a sum of terms, each of which includes a binomial coefficient, a power of a, and a power of b. This theorem is fundamental in algebra and serves as a bridge between combinatorics and polynomial algebra.

Binomial Coefficients

Binomial coefficients, typically denoted as nCk or \(\binom{n}{k}\), are numbers that quantify the number of ways to choose k elements from a set of n elements without regard to order. They are essential in determining the coefficients in the expansion of (a + b)^n and can be computed using factorials or found directly from Pascal's Triangle.

General Term in Binomial Expansion

The general term of the expansion of (a + b)^n is expressed as \(\binom{n}{k} a^{n-k}b^k\), where k runs from 0 to n. This formula allows for the calculation of any specific term in the expansion by substituting the appropriate values for n and k, ensuring a systematic approach to the expansion process.

Properties of Exponents

Understanding the laws of exponents is crucial when expanding binomials. These properties, such as the product of powers and power of a product rules, ensure that expressions with variables and constants are correctly raised to the required powers. This is particularly important in correctly managing the distribution of the exponent across the terms in the binomial expansion.

Transcript

00:01 If we want to expand the binomial 2x plus 3 to the 5th, we can use the binomial theorem.

00:12 So in this case, x over here equals 2x in our binomial.

00:19 A equals 3, and n over here equals 5.

00:26 So using the binomial theorem for the first term, j equals 0, which is specified here.

00:33 So n equals 5, so 5 to 0 times 2x to the n minus j, so 5 minus 0, which is still 5, times a, which is 3 to the j, which is 0.

00:56 The second term, j equals 1.

00:58 So it's 5 choose 1, times 2x to the 5 minus 1, which is 4 times 3 to the 1.

01:06 For the third term, j equals 2.

01:10 So it's 5 choose 2 times 2x to the 5 minus 2, which is 3, times 3 to the 2 to the 2 times 3 to the 2 times 3 to the 2.

01:19 Next term, j equals 4.

01:23 So you have 5 choose 4 times 2x to the 5 minus 4, which is 1, times 3 to the 4 times 3 to the 4.

01:45 Now for the last term, j equals 5.

01:49 So you have 5 choose 5 times 2x to the 5 minus 5, so to the 0 power, times 3 to the 5th.

01:59 Now all that's left to do is condense this a little bit.

02:04 Now, each of these 5 to 0 all the way up to 5 choose 5, you can do by hand.

02:10 Or you can look at pascal's triangle using the fifth row because this is a fifth degree binomial.

02:23 So 5 to 0 equals 1...

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