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Chapter 12: Problem 19

How many terms are there in the expansion of \((x+y)^{8} ?\)

Short Answer

Expert verified
There are 9 terms in the expansion.

Step by step solution

01

Understanding the Problem

We need to determine the number of terms in the expansion of \((x+y)^8\). This is a binomial expression raised to a power of 8.
02

Applying the Binomial Theorem

The binomial theorem states that \((x+y)^n\) expands into \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Therefore, the expansion will have terms for each \(k\) from 0 to \(n\).
03

Counting the Terms

The number of terms in the expansion corresponds to the number of different powers \(k\) can take, which is from 0 to \(n\). Thus, there are \(n+1\) terms.
04

Calculating the Number of Terms

Since \(n=8\) in \((x+y)^8\), plug this into the formula from Step 3. Therefore, the number of terms is \(8+1=9\).

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful mathematical tool that allows us to expand expressions that are in the form of a binomial, such as \((a + b)^n\). It provides a formula that breaks down these expressions into separate terms, making calculations much simpler.
For \((x + y)^n\), the binomial theorem states:
  • Each term in the expansion is determined by the binomial coefficients \(\binom{n}{k}\), which are also known as "combinations."
  • The expression \((x + y)^n\) expands to: \[\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
  • Every term corresponds to a different value of \(k\), from 0 up to \(n\).
These coefficients, \(\binom{n}{k}\), count the number of ways to choose \(k\) items from \(n\), mathematically expressed as \(\frac{n!}{k!(n-k)!}\). Using the theorem not only aids in expanding binomials but also in calculating probabilities.
Polynomial Expansion
Polynomial Expansion involves breaking down expressions into sums of simpler powers. When we speak about the polynomial expansion of \((x+y)^8\), it refers to expressing it as a sum of terms involving powers of \(x\) and \(y\).
Here's what happens:
  • In \((x+y)^8\), starting from \( (x+y)^1 = x + y\), multiplying the expression by itself repeatedly generates terms like \(x^8, \; x^7y, \; x^6y^2,\) and so forth.
  • Each term has a format where the exponents on \(x\) and \(y\) add up to 8, for example \(7+1,\; 6+2,\; \ldots\).
This expansion is what gives us insight into the number of terms (or distinct combinations of the powers of \(x\) and \(y\)), which is \(n+1\) as derived from counting terms with differing powers of \(y\), ranging from 0 to \(n\). It鈥檚 a methodical approach that鈥檚 crucial not only in algebra but also in calculus and other mathematical fields.
Combinatorics
Combinatorics, the branch of mathematics dealing with counting and arrangements, plays a central role in understanding polynomial expansions through the binomial theorem.
Why is it important?
  • It helps us determine how many terms the expansion of \((x+y)^n\) has by using the "combinations" principle, seen in the theorem's coefficients.
  • For instance, in \((x+y)^8\), using combinatorics, we calculate that there are 9 different combinations or terms, which correspond to the number of ways we can pick \(k\) elements from 8 (ranging \(k\) from 0 to 8).
  • Combinatorics not only aids in expansions but also answers broader questions involving arrangements, such as determining possible outcomes in probability.
  • The concept of \(\binom{n}{k}\) becomes more than just a term; it鈥檚 about understanding the possibilities and choosing functions within any set.
This branch simplifies otherwise complex problems and has applications ranging from basic arithmetic to advanced statistical models. In other words, understanding combinatorics can give creative solutions to everyday problems.

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Most popular questions from this chapter

The average family in the United States spends 150 dollars on food per week. Write a general term \(a_{n}\) for a sequence that gives the spending on food after \(n\) weeks. Find \(a_{4}\) and interpret the result.

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{31}=5580, a_{31}=360$$

Find the sum of each series. $$\sum_{i=1}^{5}(i-8)$$

Find the sum of each series. $$\sum_{i=1}^{12}(-5-8 i)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{12}=-108, a_{12}=-19$$
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