/*! 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 13 \(\frac{-2 s^{2}-3 s-2}{s(s+1)^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 13

\(\frac{-2 s^{2}-3 s-2}{s(s+1)^{2}}\)

Short Answer

Expert verified
The simplified form of the rational function is \(\frac {-(s + 2)(2s + 1)}{s(s+1)^{2}}\). There are no factors to cancel in the numerator and denominator.

Step by step solution

01

Factorize the Numerator

Start by expressing the numerator of the fraction as a product of its factors. Hence, \(-2 s^{2}-3 s-2\) can be factored into \(-(s + 2)(2s + 1)\).
02

Rewrite the Function Using the Factored Form

Rewrite the function using the factored form of the numerator. The rational function will be \(\frac {-(s + 2)(2s + 1)}{s(s+1)^{2}}\).
03

Perform Cancelations

There are no factors common to both the numerator and denominator of our current fraction, hence no further simplification can be achieved.

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!

Key Concepts

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

Factoring Polynomials
Factoring polynomials might sound tricky, but it becomes straightforward with practice. The goal is to rewrite a polynomial as a product of simpler polynomials. For instance, consider the quadratic polynomial \[-2s^2 - 3s - 2\].
A good starting point is identifying patterns or using the quadratic formula.
  • Look for two numbers that multiply to give the product of the first and last coefficients (-2 and -2) and add up to the middle coefficient (-3).
  • In our example, \(-1\) and \(-2\) work, which allows us to factor the quadratic as \(-(s + 2)(2s + 1)\).
Factoring breaks down complex expressions and is crucial for further simplification or solving equations.
Rational Functions
Rational functions are fractions where the numerator and denominator are polynomials. Understanding these functions is key to solving a variety of algebraic problems. For example, \(\frac{-2s^2 - 3s - 2}{s(s+1)^2}\) is a rational function.
  • The numerator is the polynomial \(-2s^2 - 3s - 2\).
  • The denominator is the polynomial \(s(s+1)^2\).
To work with rational functions effectively, you should:
  • Factor both the numerator and denominator fully.
  • Identify and cancel common factors (if any).
While no cancellations were possible here, keeping this approach in mind helps in simplifying such functions in general.
Algebraic Simplification
Simplifying algebraic expressions makes them easier to work with. It's like tidy up math! To simplify a rational function, you should factor and then cancel common terms if possible.
In our example, after factoring both the numerator and denominator, we saw:
  • Numerator: \(-(s + 2)(2s + 1)\)
  • Denominator: \(s(s+1)^2\)
With no common factors to cancel, the expression is as simplified as it gets. Simplification reduces complexity and allows for easier evaluation or integration of functions later on. Though no further reduction was possible here, always re-evaluate for further steps when dealing with real-world math problems.

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

\(\int_{-\infty}^{\infty}(\sin 3 t) \delta\left(t-\frac{\pi}{2}\right) d t\)

$$t^{3} \delta(t-3)$$

27\. $$y^{\prime \prime}-2 y^{\prime}+5 y=g(t) ; \quad y(0)=0, \quad y^{\prime}(0)=2$$

$$\begin{array}{l}{y^{\prime \prime}-y=4 \delta(t-2)+t^{2}} \\ {y(0)=0, \quad y^{\prime}(0)=2}\end{array}$$

$$w^{\prime \prime}+w=\delta(t-\pi) ; \quad w(0)=0, \quad w^{\prime}(0)=0$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

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.