/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 With a graphing utility, plot \(... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 62

With a graphing utility, plot \(y_{1}=(x+3)^{4}\) \(y_{2}=x^{4}+4 x^{3}+6 x^{2}+4 x+1,\) and \(y_{3}=x^{4}+12 x^{3}+54 x^{2}+108 x+81 .\) Which is the binomial expansion of \((x+3)^{4}, y_{2}\) or \(y_{3} ?\)

Short Answer

Expert verified
The binomial expansion of \((x+3)^4\) is \(y_3\).

Step by step solution

01

Understand the problem

We are given three functions, and we need to determine which one is the binomial expansion of \((x + 3)^4\). The functions are \(y_1 = (x+3)^4\), \(y_2 = x^4 + 4x^3 + 6x^2 + 4x + 1\), and \(y_3 = x^4 + 12x^3 + 54x^2 + 108x + 81\).
02

Recall Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = x\), \(b = 3\), and \(n = 4\).
03

Expand \((x+3)^4\) Using Binomial Theorem

Using the binomial theorem for \((x+3)^4\):\((x+3)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} 3^k\).Calculate each term:- \(k=0\): \(\binom{4}{0} x^4 3^0 = x^4\)- \(k=1\): \(\binom{4}{1} x^3 3^1 = 4x^3 \cdot 3 = 12x^3\)- \(k=2\): \(\binom{4}{2} x^2 3^2 = 6x^2 \cdot 9 = 54x^2\)- \(k=3\): \(\binom{4}{3} x 3^3 = 4x \cdot 27 = 108x\)- \(k=4\): \(\binom{4}{4} 3^4 = 81\)Thus, \((x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81\).
04

Compare Expansions

Compare the derived binomial expansion \(x^4 + 12x^3 + 54x^2 + 108x + 81\) with \(y_2\) and \(y_3\). \(y_3 = x^4 + 12x^3 + 54x^2 + 108x + 81\) matches the expansion, confirming that \(y_3\) is the binomial expansion of \((x+3)^4\).

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Key Concepts

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

Graphing Utility
A graphing utility is a powerful tool designed to assist you in visualizing mathematical functions and equations. Such tools can be physical, like a graphing calculator, or digital, like online math software and apps. Here's why they're handy:
  • They provide a visual representation of abstract mathematical concepts.
  • They allow for easy comparison of functions by plotting multiple graphs simultaneously.
  • You can quickly adjust parameters to see how changes affect the graph.
To plot the functions given in the exercise, you enter each equation into the graphing utility. The graphing utility will display shapes and lines that help identify features such as behavior, intersections, or overlaps of the functions. This kind of visual aid makes it easier to visually compare the functions, like assessing whether a function matches a particular polynomial expansion.
Binomial Theorem
The Binomial Theorem is a fundamental principle in algebra that provides a formula for expanding expressions that are raised to a power. Its utility comes from simplifying complex expressions. The theorem is expressed as: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here's what each part means:
  • \(a\) and \(b\) are terms in a binomial expression.
  • \(n\) is the power to which the binomial is raised.
  • \(\binom{n}{k}\) is a binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).
In the given problem, substituting \(a = x\), \(b = 3\), and \(n = 4\) into the theorem, results in a sequence of terms that gives the expanded form. This expansion helps you determine which function models the expansion of \((x+3)^4\) correctly. Understanding each step in the expansion is crucial in comparing the function to its possible polynomial forms.
Polynomial Functions
Polynomial functions are expressions involving sums of powers of variables, typically in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0\). Understanding the structure of polynomials helps in recognizing and comparing expressions like those in the exercise. Key features include:
  • Degree: The highest power of the variable \(x\) in the polynomial, indicating its behavior as \(x\) approaches infinity.
  • Coefficients: Constants multiplied by each power of the variable, influencing shape and orientation of the graph.
  • Roots and Intercepts: Points where the polynomial equals zero or crosses axes, important for graphing utility checks.
By examining the polynomial form obtained through the Binomial Theorem, you match it against provided forms. Recognizing the correct polynomial describes the same function as its binomial counterpart, ensuring the relationship between the algebraic expression and its visual graph remains consistent and clear.

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Most popular questions from this chapter

In calculus, we study the convergence of sequences. A sequence is convergent when its terms approach a limiting value. For example, \(a_{n}=\frac{1}{n}\) is convergent because its terms approach zero. If the terms of a sequence satisfy \(a_{1} \leq a_{2} \leq a_{3} \leq \ldots \leq a_{n} \leq \ldots\) the sequence is monotonic nondecreasing. If \(a_{1} \geq a_{2} \geq a_{3} \geq \ldots \geq a_{n} \geq \ldots,\) the sequence is monotonic nonincreasing. Classify each sequence as monotonic or not monotonic. If the sequence is monotonic, determine whether it is nondecreasing or nonincreasing. $$a_{n}=\sin \left(\frac{n \pi}{4}\right)$$

Evaluate each finite series. $$\sum_{k=0}^{4} \frac{(-1)^{k} x^{k}}{k !}$$

Simplify each ratio of factorials. $$\frac{32 !}{30 !}$$

In calculus, when estimating certain integrals, we use sums of the form \(\sum_{i=1}^{n} f\left(x_{i}\right) \Delta x,\) where \(f\) is a function and \(\Delta x\) is a constant. Find the indicated sum. $$\sum_{i=1}^{100} f\left(x_{i}\right) \Delta x, \text { where } f\left(x_{i}\right)=2 i \text { and } \Delta x=0.1$$

A college student tries to save money by bringing a bag lunch instead of eating out. He will be able to save 100 dollars per month. He puts the money into his savings account, which draws \(1.2 \%\) interest and is compounded monthly. The balance in his account after \(n\) months of bagging his lunch is $$A_{n}=100,000\left[(1.001)^{n}-1\right] \quad n=1,2, \ldots$$ Calculate the first four terms of this sequence. Calculate the amount after 3 years ( 36 months).
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