Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 7: Problem 2

Give an example of each of the following. a. A simple linear factor b. A repeated linear factor c. A simple irreducible quadratic factor d. A repeated irreducible quadratic factor

Short Answer

Expert verified
Question: Provide an example of each type of factor and explain its properties. Answer: (a) A simple linear factor is (x - 3). It has a single root at x = 3. (b) A repeated linear factor is (x - 2)^2. It has a root at x = 2 with multiplicity 2. (c) A simple irreducible quadratic factor is x^2 + 1. It has no real roots, as the discriminant is -3 < 0. (d) A repeated irreducible quadratic factor is (x^2 + 1)^2. It has no real roots, and it has a multiplicity of 2.

Step by step solution

01

(a) Simple Linear Factor

A simple linear factor is a linear expression with a single root. It has the general form \((x - a)\), where \(a\) is a constant. An example of a simple linear factor is \((x - 3)\). This factor has a single root at \(x = 3\).
02

(b) Repeated Linear Factor

A repeated linear factor is a linear expression raised to a certain power greater than 1. It has the general form \((x - a)^n\), where \(a\) is a constant and \(n\) is an integer greater than 1. An example of a repeated linear factor is \((x - 2)^2\). This factor has a root at \(x = 2\) with multiplicity 2.
03

(c) Simple Irreducible Quadratic Factor

A simple irreducible quadratic factor is a quadratic expression that cannot be factored further and has no real roots. It has the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and the discriminant \(b^2 - 4ac < 0\). An example of a simple irreducible quadratic factor is \(x^2 + 1\). This factor has no real roots, as the discriminant is \((-1)^2 - 4(1)(1) = -3 < 0\).
04

(d) Repeated Irreducible Quadratic Factor

A repeated irreducible quadratic factor is an irreducible quadratic expression raised to a certain power greater than 1. It has the general form \((ax^2 + bx + c)^n\), where \(a\), \(b\), and \(c\) are constants, the discriminant \(b^2 - 4ac < 0\), and \(n\) is an integer greater than 1. An example of a repeated irreducible quadratic factor is \((x^2 + 1)^2\). This factor has no real roots, as the discriminant is \((-1)^2 - 4(1)(1) = -3 < 0\), and it has a multiplicity of 2.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Let \(I_{n}=\int x^{n} e^{-x^{2}} d x,\) where \(n\) is a nonnegative integer. a. \(I_{0}=\int e^{-x^{2}} d x\) cannot be expressed in terms of elementary functions. Evaluate \(I_{1}\). b. Use integration by parts to evaluate \(I_{3}\). c. Use integration by parts and the result of part (b) to evaluate \(I_{5}\). d. Show that, in general, if \(n\) is odd, then \(I_{n}=-\frac{1}{2} e^{-x^{2}} p_{n-1}(x)\) where \(p_{n-1}\) is a polynomial of degree \(n-1\). e. Argue that if \(n\) is even, then \(I_{n}\) cannot be expressed in terms of elementary functions.

The cycloid is the curve traced by a point on the rim of a rolling wheel. Imagine a wire shaped like an inverted cycloid (see figure). A bead sliding down this wire without friction has some remarkable properties. Among all wire shapes, the cycloid is the shape that produces the fastest descent time. It can be shown that the descent time between any two points \(0 \leq a \leq b \leq \pi\) on the curve is $$\text { descent time }=\int_{a}^{b} \sqrt{\frac{1-\cos t}{g(\cos a-\cos t)}} d t$$ where \(g\) is the acceleration due to gravity, \(t=0\) corresponds to the top of the wire, and \(t=\pi\) corresponds to the lowest point on the wire. a. Find the descent time on the interval \([a, b]\) by making the substitution \(u=\cos t\) b. Show that when \(b=\pi\), the descent time is the same for all values of \(a ;\) that is, the descent time to the bottom of the wire is the same for all starting points.

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ Verify relation \(A\) by differentiating \(x=2 \tan ^{-1} u\). Verify relations \(B\) and \(C\) using a right-triangle diagram and the double-angle formulas $$\sin x=2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \text { and } \cos x=2 \cos ^{2}\left(\frac{x}{2}\right)-1$$

Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \sin 5 x \sin 7 x d x$$

On the interval \([0,2],\) the graphs of \(f(x)=x^{2} / 3\) and \(g(x)=x^{2}\left(9-x^{2}\right)^{-1 / 2}\) have similar shapes. a. Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,2] b. Find the area of the region bounded by the graph of \(g\) and the \(x\) -axis on the interval [0,2] c. Which region has the greater area?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.

Sign-up for free