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Chapter 8: Problem 35

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{16} $$

Short Answer

Expert verified
The first three terms of the binomial expansion are \(x^{32}\), \(16x^{30}\), and \(120x^{28}\).

Step by step solution

01

Identify the terms of the binomial

Recognize that the binomial is \(x^2 + 1\), and power is 16 in binomial expansion.
02

Calculate first term of the binomial expansion

According to the binomial theorem, the first term is \((x^2)^{16}\) which simplifies to \(x^{32}\)
03

Calculate second term of the binomial expansion

The second term is given by \((16C1)*(x^2)^{15}*(1)\), which simplifies to \(16*x^{30}\)
04

Calculate third term of the binomial expansion

The third term is given by \((16C2)*(x^2)^{14}*(1)\), which simplifies to \(120*x^{28}\)

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Key Concepts

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

Binomial Theorem
When we raise a binomial鈥攁n algebraic expression containing two terms鈥攖o a power, we use the Binomial Theorem to expand it into a series of terms. This theorem is a fundamental concept in algebra that simplifies the process of expanding polynomials.

According to the Binomial Theorem, the expansion of \( (a + b)^n \) involves terms with coefficients known as binomial coefficients. These coefficients can be determined using combinations. For the expansion of \( (x^2 + 1)^{16} \) that we're interested in, the Binomial Theorem helps us to find the first three terms without having to multiply the binomial sixteen times by itself.
Binomial Coefficients
Binomial coefficients are numerical factors that determine the number of ways we can choose elements from a larger set. They are an integral part of the Binomial Theorem. Represented as \( nCk \) or \( \binom{n}{k} \), they correspond to the coefficients in the expansion of a binomial raised to a power.

For instance, the second term of our binomial expansion \( (x^2 + 1)^{16} \) involved the coefficient \( 16C1 \) which is 16, because there are 16 different ways to choose 1 term out of 16. Similarly, \( 16C2 \) for our third term is 120, indicating the number of ways to choose 2 terms out of 16.
Polynomial Simplification
Polynomial simplification involves reducing the complexity of a polynomial expression by combining like terms and using mathematical operations properly. It leads to a more compact and comprehensible mathematical expression.

While expanding a binomial, polynomial simplification plays a crucial role. After writing down the terms of the binomial expansion with their respective binomial coefficients, each term must be simplified by raising the binomial terms to their appropriate powers and multiplying out. In our exercise, simplifying \( (x^2)^{15} \) as part of the second term resulted in \( x^{30} \) following the rules of exponents.
Algebraic Expressions
Algebraic expressions are combinations of constants, variables, and arithmetic operations. In our current problem, \( x^2 + 1 \) is a binomial algebraic expression with two terms. Understanding how to manipulate these expressions through addition, subtraction, multiplication, and exponentiation is essential in algebra.

Each term in the binomial expansion represents an algebraic expression derived from applying the Binomial Theorem. The expression's simplification results in the power being distributed to both variables and constants, creating a new, larger algebraic expression that represents the expanded polynomial.

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Most popular questions from this chapter

Research and present a group report on state lotteries. Include answers to some or all of the following questions: Which states do not have lotteries? Why not? How much is spent per capital on lotteries? What are some of the lottery games? What is the probability of winning top prize in these games? What income groups spend the greatest amount of money on lotteries? If your state has a lottery, what does it do with the money it makes? Is the way the money is spent what was promised when the lottery first began?

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$ \begin{array}{l}f_{1}(x)=(x+1)^{4} \\\f_{2}(x)=x^{4} \\\f_{3}(x)=x^{4}+4 x^{3} \\\f_{4}(x)=x^{4}+4 x^{3}+6 x^{2} \\\f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x \\\f_{6}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1\end{array} $$ Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

The probability that South Florida will be hit by a major hurricane (category 4 or 5) in any single year is \(\frac{1}{16}\). (Source: National Hurricane Center) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years?c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

Describe the pattern on the exponents on \(b\) in the expansion of \((a+b)^{n}\).

You are now 25 years old and would like to retire at age 55 with a retirement fund of 1,000,000 dollar. How much should you deposit at the end of each month for the next 30 years in an IRA paying \(10 \%\) annual interest compounded monthly to achieve your goal? Round to the nearest dollar.
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