/*! 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 55 Suppose \(M\) and \(N\) are odd ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 55

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no rational zeros.

Short Answer

Expert verified
The quadratic equation \(x^2 + Mx + N\) has no rational zeros because, considering that M and N are odd integers, any possible rational zero in the form p/1 (where p is a factor of N) would lead to the equation resulting in an odd integer. By the Rational Root Theorem, there are no rational zeros that would make the equation equal to zero.

Step by step solution

01

Identify the coefficients of the quadratic equation

In the given quadratic equation: \(x^2 + Mx + N\) The coefficients are: - Leading Coefficient: 1 (the coefficient of the \(x^2\)-term) - Middle Coefficient: M (the coefficient of the x-term, an odd integer) - Constant Coefficient: N (an odd integer)
02

Apply the Rational Root Theorem

By the Rational Root Theorem, any rational zero of this equation can be expressed as p/q, where p is a factor of N and q is a factor of the leading coefficient, which is 1. Thus, the possible rational zeros will be of the form p/1, where p is a factor of N.
03

Evaluate the quadratic equation for possible rational zeros

Let p be a factor of N and substitute \(\frac{p}{1}\), or simply p, into the quadratic equation: \[(p)^2 + M(p) + N\] Let's analyze the parity (evenness or oddness) of each term: - \(p^2\): Since p is a factor of N (an odd integer), it must be odd as well. An odd times an odd is still odd, so this term is odd. - \(Mp\): M is an odd integer, and p is also odd (as stated before). An odd times an odd is again odd, making this term odd as well. - N: It's given that N is an odd integer.
04

Determine the parity of the sum of all terms of the evaluated equation

Now let's sum up all the terms: Odd + Odd + Odd Since an odd plus an odd results in an even, our expression simplifies to: Even + Odd An even number plus an odd number always results in an odd number. Thus, the result of the quadratic equation for any possible rational zero will be an odd integer.
05

Conclude whether the quadratic equation has any rational zeros

Since any possible rational zero of the given quadratic equation evaluates to an odd integer, it means that there are no rational zeros that would make the equation equal to zero. Therefore, we can conclude that the quadratic equation \(x^2 + Mx + N\) has no rational zeros.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} t(x) $$

Write the indicated expression as \(a\) polynomial. $$ \frac{q(2+x)-q(2)}{x} $$

A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}+3 x^{3}+1}{x^{2}+2 x+5} $$

Give an example of a polynomial \(p\) of degree 8 such that \(p(2)=3\) and \(p(x) \geq 3\) for all real numbers \(x\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.