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Chapter 12: Problem 228

Find the indicated term in the expansion of the binomial. Fourth term of \((x-y)^{8}\)

Short Answer

Expert verified
-56x^{5}y^{3}

Step by step solution

01

Understand the Binomial Theorem

The Binomial Theorem states: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In this case, we need to find the fourth term of \((x - y)^{8}\).
02

Identify the General Term

The general term in the expansion is given by\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\]For \((x - y)^{8}\), \(a = x\), \(b = -y\), and \(n = 8\).
03

Substitute Values for the Fourth Term

The fourth term corresponds to \(k = 3\) (since we start indexing at 0). Substituting in the general term formula:\[T_{4} = \binom{8}{3} x^{8-3} (-y)^3\]Calculate the power terms: \(x^{5}\) and \((-y)^3 = -y^3\).
04

Calculate the Binomial Coefficient

Calculate \(\binom{8}{3}\):\[\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = 56\]
05

Put It All Together

Now, put everything together to get the fourth term:\[T_{4} = 56 x^{5} (-y)^{3} = 56 x^{5} (-y^{3}) = -56 x^{5} y^{3}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion
Binomial expansion is a fundamental concept in algebra. When you see an expression like \((a + b)^n\), the Binomial Theorem helps us expand it into a sum containing terms of the form \binom{n}{k} a^{n-k} b^k\. This means we break it down into simpler pieces that are each a product of a binomial coefficient and power terms. For example, in the exercise given, we need to find a specific term in the expansion of \((x - y)^8\). By using binomial expansion, we can systematically determine every term in this expanded form.
Binomial Coefficient
The binomial coefficient is an integral part of the binomial expansion. It's represented as \binom{n}{k}\ and is sometimes read as 'n choose k'. It tells us how many different ways we can pick \k\ items from \ items without considering the order. Mathematically, it's defined as \binom{n}{k} = \frac{n!}{k!(n-k)!}\. In the context of the exercise, for the fourth term of \((x-y)^8\), we identified \binom{8}{3}\ because we need the index \k = 3\ to find that specific term. We calculated \binom{8}{3} = 56\, revealing how many multiples of this term's power combinations appear in the expanded form.
Power Terms
Power terms refer to the exponents applied to the variables in our binomial expansion. Each term in the binomial expansion will have a product of powers of \a\ and \b\. Specifically, let's consider our exercise: the fourth term of \((x - y)^8\). The general term is represented as \T_{k+1} = \binom{n}{k} a^{n-k} b^k\. Here, for \k = 3\, we get \x^{5}\ and \-y^{3}\. Notice how the power of \x\ decreases while the power of \-y\ increases as we move along each term of the expansion. These exponents, combined with binomial coefficients, construct each unique term in the expansion.

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Most popular questions from this chapter

Find the sum of the first 50 terms of the arithmetic sequence whose general term is given. \(a_{n}=-3 n+5\)

Find the sum of each infinite geometric series. \(6-2+\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+\ldots\)

Write the first five terms of each geometric sequence with the given first term and common ratio. \(a_{1}=9\) and \(r=2\)

Find the sum of each infinite geometric series. \(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\frac{1}{729}+\ldots\)

Determine if the sequence is geometric, and if so, indicate the common ratio. \(3,12,48,192,768,3072, \ldots\)
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