Use Pascal's triangle to expand the binomial. (q-r)^7 |...

Name: Problem 6, Chapter 14: Beginning and Intermediate Algebra
Uploaded: 2020-04-09T19:02:15-08:00
Duration: 4 min
Channel: Julie Silva
Description: Use Pascal's triangle to expand the binomial. (q-r)^7

Key Concepts

Binomial Theorem

The binomial theorem provides a formula for expanding expressions raised to a power, such as (a + b)^n. It expresses the expansion as a sum of terms involving binomial coefficients and the corresponding powers of the two terms. This theorem is fundamental in algebra for simplifying and computing power expressions.

Pascal's Triangle

Pascal's Triangle is a geometric arrangement of numbers where each number is the sum of the two directly above it. Its rows correspond to the binomial coefficients used in the binomial theorem, providing an easy and structured way to determine the coefficients for each term in a binomial expansion.

Binomial Coefficients

Binomial coefficients represent the number of ways to choose a subset of elements from a larger set and are central to the binomial expansion. They appear as the coefficients in the expanded form of a binomial expression and are directly obtainable from Pascal's Triangle, reflecting the combinatorial aspect of the expansion.

Algebraic Expansion with Negative Terms

When expanding a binomial that involves a negative term, such as (q - r)^n, it is important to account for the alternating signs that result from raising a negative number to successive powers. This adjustment is crucial to correctly expand and simplify the expression while maintaining the proper signs for each term.

Transcript

00:01 We're being asked to use pascal's triangle to expand q minus r to the 7th power.

00:05 Well, because the exponent is 7, we have to use the n -equal 7 row in pascal's triangle, which is 1, 7, 21, 35, 35, 21, 7, and 1.

00:21 So as we can see, we're going to have 8 terms.

00:23 So let's get started.

00:25 Well, the first terms coefficient will be 1.

00:27 Then we start with our first term, which is q, and it's getting raised to the 7th power.

00:32 And our second term is negative r, and it's going to get raised to the 0 power.

00:36 Remember, our exponent should always add up to our exponent, which in this case is 7.

00:41 So now the coefficient of our second term is 7.

00:44 So we're going to decrease the exponent of q by 1, so q to the 6, and increase the exponent for negative r by 1.

00:51 So that'll be negative r to the 1.

00:53 The next coefficient is 21, and then we'll have q to the 5th because we drop it by 1, and we have negative r to the second because we add 1.

01:02 The next coefficient will be 35, and then we'll have q to the 4th because we drop the exponent by 1, and we'll have negative r to the 3rd because we add 1.

01:13 The next coefficient will also be 35.

01:16 Q is now going to raise to the 3rd because we decrease the exponent by 1, and negative r will have an exponent of 4 because we increase this exponent by 1.

01:26 The next term, we'll have a coefficient of 21, and then we'll have q squared because we decreased by 1, and negative r to the 5th, because we increased by 1.

01:36 And then our next term, i'm going to come down here, because i'm running out of room, we'll have a coefficient of 7, and then we'll have q to the 1 because we decreased it by 1, and negative r to the 6 because we increased it by 1.

01:48 And lastly, our last term will have a coefficient of 1, q will be raised to the 0, because we decrease the exponent by 1, and negative r will be raised to the 7 because we increased it by 1.

02:00 Well, now we just need to go ahead and simplify this.

02:04 So let's start with that first term.

02:06 We have 1 q to the 7th times negative r to the 0...

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