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Chapter 9: Problem 33

Expanding an Expression In Exercises \(19-40,\) use the Binomial Theorem to expand and simplify the expression. $$\left(x^{2}+v^{2}\right)^{4}$$

Short Answer

Expert verified
\((x^2 + v^2)^4 = (x^2)^4 + 4(x^2)^3(v^2) + 6(x^2)^2(v^2)^2 + 4(x^2)(v^2)^3 + (v^2)^4 = x^8 + 4x^6v^2 + 6x^4v^4 + 4x^2v^6 + v^8 \)

Step by step solution

01

Identifying the Elements

Identify the elements in the given expression. Here, 'a' represents \(x^2\), 'b' represents \(v^2\) and 'n' represents 4 in the Binomial Theorem.
02

Applying the Binomial Theorem

Apply the Binomial Theorem formula, \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}\). Binomial coefficients (the part \(\binom{n}{k}\)) can be calculated using the formula \(\frac{n!}{(n-k)! k!}\), where '!' represents factorial, or can be taken from Pascal's Triangle for n = 4.
03

Expanding The Expression

Substitute the values of a, b and n into the binomial theorem formula and simplify by expanding the expression. This involves - term 1: when k=0, term 2: when k=1, term 3: when k=2, term 4: when k=3, term 5: when k=4.
04

Simplifying The Expansion

Combine the like terms in the expanded expression to simplify it. In this case, there will be no like terms, hence the simplified form will be the sum of all the terms (1st to 5th term).

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Key Concepts

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

Expansion of Expressions
When we talk about the expansion of expressions using the Binomial Theorem, we refer to breaking down a complex expression into simpler parts. This is especially useful when dealing with powers of a binomial expression like \((x^2 + v^2)^4\). Expanding such expressions involves rewriting them as a series of terms...
  • The terms are based on different values of 'k' from 0 up to 'n', which in this case is 4.
  • Each term involves a combination of the powers of the individual elements \(x^2\) and \(v^2\) raised to different exponents.
The process of expansion not only simplifies complex expressions but also makes it easier to identify patterns and relationships within mathematical expressions. By following the Binomial Theorem, which systematically lays out the terms, one can turn an expression like \((x^2 + v^2)^4\) into a more manageable form.
Binomial Coefficients
In the expansion of binomial expressions, you'll often come across binomial coefficients, which are crucial for determining the weight of each term. These coefficients, denoted by \(\binom{n}{k}\), represent the number of ways to choose 'k' elements from a set of 'n' elements.
  • They are integral to forming the expanded expression and can be calculated using the formula \(\frac{n!}{(n-k)! k!}\).
  • Alternatively, binomial coefficients can be quickly identified using Pascal's Triangle.
For an example involving \((x^2 + v^2)^4\), the coefficients for the terms when 'k' ranges from 0 to 4 would be derived from the row corresponding to 'n' in Pascal's Triangle, making this process much easier and visual.
Factorials
A factorial, denoted by an exclamation mark (!), is a mathematical operation that multiplies a sequence of descending natural numbers. It is a core component in the calculation of binomial coefficients.
  • For example, \(n!\) means multiplying all whole numbers from 'n' down to 1.
  • Factorials are used to calculate the number of ways to arrange 'n' items, which directly relates to combinations.
Understanding factorials is critical because they allow you to determine how binomial coefficients are computed in expressions such as \(\binom{n}{k} = \frac{n!}{(n-k)!k!}\). This understanding provides a concrete foundation for using the Binomial Theorem in expanding binomials.

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Most popular questions from this chapter

Expanding a Complex Number In Exercises \(73-78\) , use the Binomial Theorem to expand the complex number. Simplify your result. $$(5+\sqrt{-9})^{3}$$

Proof In Exercises \(99-102,\) prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n .\) The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\) .

Expanding an Expression In Exercises \(61-66,\) use the Binomial Theorem to expand and simplify the expression. $$\left(u^{3 / 5}+2\right)^{5}$$

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{i=1}^{6}\left(6 i-8 i^{3}\right)$$

Consider a group of \(n\) people. (a) Explain why the following pattern gives the probabilities that the \(n\) people have distinct birthdays. $$\begin{array}{l}{n=2 : \frac{365}{365} \cdot \frac{364}{365}=\frac{365 \cdot 364}{365^{2}}} \\ {n=3 : \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 \cdot 364 \cdot 363}{365^{3}}}\end{array}$$ (b) Use the pattern in part (a) to write an expression for the probability that \(n=4\) people have distinct birthdays. (c) Let \(P_{n}\) be the probability that the \(n\) people have distinct birthdays. Verify that this probability can be obtained recursively by $$P_{1}=1\( and \)P_{n}=\frac{365-(n-1)}{365} P_{n-1}$$ (d) Explain why \(Q_{n}=1-P_{n}\) gives the probability that at least two people in a group of \(n\) people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than \(\frac{1}{2} ?\) Explain.
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