Expand each expression using the Binomial Theorem. (x^2-y^2)^6

Name: Problem 24, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-05-14T13:35:51-08:00
Duration: 5 min 35 s
Channel: John Vanschoick
Description: Expand each expression using the Binomial Theorem. (x^2-y^2)^6

Expand each expression using the Binomial Theorem....

Key Concepts

Binomial Theorem

A fundamental theorem in algebra that provides a formula for expanding expressions raised to a power in the form (a + b)^n, where the expansion is expressed as a sum involving terms with binomial coefficients. This theorem simplifies the process of expansion by breaking it down into a systematic combination of powers of a and b.

Binomial Coefficients

These coefficients, typically denoted as 'n choose k', represent the number of ways to choose k elements from a set of n elements and are calculated using factorials. They are central to the binomial expansion as they determine the weight of each term in the expanded form.

Exponentiation Rules

Rules governing the manipulation of expressions with exponents, such as the power of a power rule which states that (x^m)^n equals x^(m*n). These rules are essential in correctly applying the binomial theorem, especially when the binomial terms themselves involve exponents.

Transcript

00:01 If we want to expand the binomial expression x squared minus y squared to the sixth, we can use the binomial theorem.

00:10 Before we get started, we need to rewrite this equation so it fits into this form.

00:16 So we can rewrite this as x squared plus negative y squared all to the sixth.

00:33 So now we could see that in this expression, x equals x squared, a, over here equals negative y squared and n over here equals six.

00:55 So when we write this out, for the first term, j equals zero.

01:00 So we get n, which equals six, to use j equals zero, times x squared to the n minus j, which is x squared to the sixth, times a, which is negative y squared to the j, which is zero.

01:20 The next term, j equals 1, so 6 to the 1st times x squared to the n minus j, 6 minus 1, which is 5, times negative y squared to the first.

01:38 The third term, j equals 2.

01:40 So 6 choose 2 times x squared to the 5 to the 6 minus 2, which is 4, times negative y squared to the 2 to the second.

01:56 Now j equals 3 and we get 6 choose 3 times x squared to the 6 minus 3 which is 6 times negative y squared to the 3.

02:08 Next term j equals 4 so we get 6 choose 4 times x squared to the 4 to the 6 minus 4 which is 2 times negative y squared to the 4th.

02:27 Next j equals 5 so we get 6 choose 5 times x squared to the 1st times negative y squared to the fifth.

02:40 And finally, j equals 6, and we get 6 to 6 times x squared to the 0, times negative y squared to the 6.

02:53 Now all that's left is to simplify this.

02:58 Now, for 6 to 0 all the way to 6 to 6, we can do that by hand, or we can use pascal's triangle.

03:07 Because our binomial is a sixth degree binomial, we need to use the sixth line of pascal's triangle to get these coefficients...

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