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Chapter 14: Problem 39

Find the indicated term in each expansion. $$(2 x+y)^{6} ; \text { third term }$$

Short Answer

Expert verified
The third term in the expansion of \((2x+y)^6\) is \(240x^{4}y^{2}\).

Step by step solution

01

Understand the binomial theorem and its structure for terms

The binomial theorem allows us to calculate the expansion of \((a+b)^n\) without multiplying it out. Each term in the expansion has the form \(\binom{n}{k}a^{n-k}b^k\), where \(\binom{n}{k}\) is a binomial coefficient, often read as 'n choose k'. Thus, to find the third term in the binomial expansion, we need to set \(k=2\) (since we start counting from \(k=0\)).
02

Calculation of the third term

Substitute \(n=6\), \(k=2\), \(a=2x\), and \(b=y\) into the binomial theorem. The third term \(T_{3}\) in the expansion is given by \[T_{3}=\binom{6}{2}(2x)^{6-2}(y)^2\].
03

Evaluate the binomial coefficient and simplify the term

The binomial coefficient \(\binom{6}{2}\) can be calculated as \(\frac{6!}{2!(6-2)!}=15\). Now substitute this value into the equation and simplify: \[T_3= 15*(2x)^4*(y)^2=240x^{4}y^{2}\]

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Key Concepts

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

Binomial Expansion
The Binomial Expansion is a way of expressing powers of a binomial as a sum of terms. When you have a binomial, which is an expression consisting of two terms, like \((a + b)^n\), the binomial theorem gives us a formula to expand it.
It tells us how to express \((a + b)^n\) as:
  • \(n + 1\) terms, where each term follows a specified pattern.
  • Each term has a coefficient known as the binomial coefficient, denoted by \(\binom{n}{k}\).
Putting these together, the expansion can be expressed in the form: \[\sum_{k=0}^n \binom{n}{k} a^{n-k}b^{k}\].
Understanding this method helps to expand powers of binomials quickly without tediously multiplying the terms by themselves repeatedly.
Binomial Coefficients
Binomial Coefficients are crucial for finding terms in a binomial expansion. They appear in the expression as \(\binom{n}{k}\), which reads as 'n choose k'. These coefficients determine how many different ways you can choose \(k\) elements from a total of \(n\) elements.
The calculation of binomial coefficients can be done using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}.\]
Here's a simple breakdown:
  • \(n!\) (n factorial) is the product of all positive integers up to \(n\).
  • This calculation ensures that each binomial coefficient is unique and helps balance the terms in the expansion.
By understanding how to calculate these coefficients, one can accurately determine each term's weight in the expansion.
Polynomial Terms
Polynomial Terms are the individual components of a polynomial expression. In the context of binomial expansion, these terms are formed using the binomial theorem. Each term in the expansion has a specific form: \( \binom{n}{k} a^{n-k} b^k \).
Key points about polynomial terms:
  • Each term involves both the powers of the original terms in the binomial, and a binomial coefficient.
  • The sum of the exponents of \(a\) and \(b\) in any term equals \(n\), the original power of the binomial.
  • The terms are arranged in decreasing powers of \(a\) and increasing powers of \(b\).
Understanding the composition of each term allows you to navigate through binomial expansions, alter the exponents, and compute specific terms accurately.

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Most popular questions from this chapter

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{100}{n} ; n:[0,1000,100] \text { by } a_{n}:[0,1,0.1]$$

If \(f(x)=x^{2}+2 x+3,\) find \(f(a+1)\) (Section \(8.1,\) Example 3 )

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=0}^{6}(-1)^{i}(i+1)^{2}=\sum_{i=1}^{7}(-1) j^{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give examples of two different arithmetic sequences whose fourth term, \(a_{4},\) is 10

What is the meaning of the symbol \Sigma? Give an example with your description.
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