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Chapter 7: Problem 12

12\. $$\frac{s+1}{\left(s^{2}+1\right)^{2}}$$

Short Answer

Expert verified
This is a complex exercise that uses many concepts from algebra and calculus. Unfortunately, it's impossible to simplify the given function any further without better understanding the context or having any additional conditions.

Step by step solution

01

Identify the Form

Identify the form of the given function to get a better understanding of how to simplify it. Look for special identities or factorizations to make the solution easier. The given function is a fractional function where both the numerator and denominator are polynomials.
02

Simplify the Denominator

Write out the square in the denominator. Any expression in the form (s^2 + 1)^2 can be rewritten as (s^4 + 2s^2 + 1), thus our expression becomes \( \frac{s + 1}{s^{4} + 2s^{2} + 1} \).
03

Polynomial Division

Apply polynomial long division. Divide (s^4 + 2s^2 + 1) by (s + 1). After dividing, the remainder is written on the top of the fraction and the quotient is written on the bottom.
04

Final Simplification

The final result after polynomial division is the final simplified version of the input function. This will be the most simplified form of the original function.

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Key Concepts

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

Fractional Functions
Fractional functions are expressions where both the numerator and the denominator are polynomials. In the exercise, the function given is \( \frac{s + 1}{(s^2 + 1)^2} \).
This type of function is also known as a rational function. It's like a fraction but with polynomials.
Fractional functions come up often in calculus and algebra because they can model a wide variety of situations and behaviors.
To understand fractional functions, it's essential to:
  • Analyze the polynomials involved.
  • Look for common factors or simplifications.
  • Consider the behavior as the variable tends to infinity.
These functions can often be simplified by finding expressions that cancel out or by factoring. So, it's all about making a complex function more manageable.
Polynomial Division
In this exercise, polynomial division is a crucial step. It's like long division, but instead, you're dividing polynomials.
The purpose is to simplify the fractional function \( \frac{s + 1}{s^4 + 2s^2 + 1} \) such that it's easier to work with.
Here are the basic steps for polynomial division:
  • Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
  • Multiply the entire divisor by this term and subtract from the original dividend.
  • Repeat the process with the new dividend until you've worked through the entire original polynomial.
Polynomial division often leaves a remainder. In fractional functions, this remainder plays a role in expressing the result more compactly.
Remember, like with numbers, the goal is to find how much of the divisor fits into the dividend.
Polynomial Simplification
Simplifying polynomials helps make expressions more manageable and often reveals important characteristics. In this problem, after carrying out polynomial division, simplification leads to a more concise expression of the function.
To simplify polynomial expressions, you should:
  • Fully expand polynomials when necessary, like for example \( (s^2 + 1)^2 \) becomes \( s^4 + 2s^2 + 1 \).
  • Factor both numerator and denominator if possible to find cancelable components.
  • Watch for special patterns such as the sum or difference of cubes.
The aim of simplification is to achieve the simplest form, where both calculation and interpretation get easier.
Imagine it as reducing clutter; polynomial simplification clears the equation landscape for us.

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Most popular questions from this chapter

Figure 7.29 shows a beam of length 2\(\lambda\) that is imbed- ded in a support on the left side and free on the right. The vertical deflection of the beam a distance \(x\) from the support is denoted by \(y(x) .\) If the beam has a concen- trated load L acting on it in the center of the beam, then the deflection must satisfy the symbolic boundary value problem $$E I y^{(4)}(x)=L \delta(x-\lambda)$$ $$y(0)=y^{\prime}(0)=y^{\prime \prime}(2 \lambda)=y^{\prime \prime \prime}(2 \lambda)=0$$ where \(E,\) the modulus of elasticity, and \(I,\) the moment of inertia, are constants. Find a formula for the dis- placement \(y(x)\) in terms of the constants \(\lambda, L, E,\) and I. [Hint: Let \(y^{\prime \prime}(0)=A\) and \(y^{\prime \prime \prime}(0)=B .\) First solve the fourth-order symbolic initial value problem and then use the conditions \(y^{\prime \prime}(2 \lambda)=y^{\prime \prime \prime}(2 \lambda)=0\) to determine \(A\) and \(B . ]\)

$$f ( t ) = \frac { t } { t ^ { 2 } - 1 }$$

\(\int_{-\infty}^{\infty} e^{-2 t} \delta(t+1) d t\)

$$\begin{array}{l}{y^{\prime \prime}+2 y^{\prime}-3 y=\delta(t-1)-\delta(t-2)} \\\ {y(0)=2, \quad y^{\prime}(0)=-2}\end{array}$$

\(\begin{array}{ll}{x^{\prime}-2 x+y^{\prime}=-(\cos t+4 \sin t) ;} & {x(\pi)=0} \\ {2 x+y^{\prime}+y=\sin t+3 \cos t ;} & {y(\pi)=3}\end{array}\)
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