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Chapter 8: Problem 11

Write the form of the partial-fraction decomposition. Do not solve for the constants. $$\frac{2 x^{3}-4 x^{2}+7 x+3}{\left(x^{2}+x+5\right)}$$

Short Answer

Expert verified
Perform polynomial long division, then express the quotient and the remainder over the divisor.

Step by step solution

01

Identify the Form

Examine the given rational function to determine its form. The numerator is a polynomial of degree 3 and the denominator is a polynomial of degree 2.
02

Degree Comparison

Since the degree of the numerator (3) is greater than the degree of the denominator (2), perform polynomial long division to simplify the initial expression.
03

Perform Polynomial Long Division

Divide the numerator \(2x^3 - 4x^2 + 7x + 3\) by the denominator \(x^2 + x + 5\). The long division will yield a quotient and a remainder with a degree less than the denominator.
04

Express as a Mixed Fraction

Write the division result as a sum of the quotient and the remainder over the divisor: \(Q(x) + \frac{R(x)}{x^2 + x + 5}\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder determined from the division.
05

Analyze the Remainder

Since \(R(x)\) is a polynomial of degree less than 2, where the denominator \((x^2 + x + 5)\) doesn't factor further, the partial-fraction decomposition of the original expression is complete without further breakdown of \(R(x)\).

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Key Concepts

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

Polynomial Division
Polynomial division is quite similar to long division with numbers. However, instead of digits, you deal with algebraic expressions involving variables like 'x.'
When dividing one polynomial by another, you aim to simplify the expression. You do this by dividing the highest degree terms first.
  • Determine what you need to multiply the leading term of the divisor to match the leading term of the dividend.
  • Multiply the entire divisor by this factor. Subtract the resulting expression from the dividend.
  • Repeat the process with the new polynomial formed after subtraction.

Continue this until the polynomial you're working with has a degree less than that of the divisor. The result of the division is a quotient plus a remainder over the initial divisor. This breakdown helps simplify complex rational expressions and is crucial for problems involving partial fraction decomposition.
Rational Functions
Rational functions are expressions that represent one polynomial divided by another. They have the form \( \frac{P(x)}{Q(x)} \) where both \(P(x)\) and \(Q(x)\) are polynomials.
These functions are important because they often arise in calculus and various applications.
For example, they can model phenomena in physics and economics. The main features of rational functions include:
  • They can have holes or vertical asymptotes if the denominator has factors that aren't canceled out by the numerator.
  • They can also have horizontal asymptotes or oblique asymptotes if the degree of the numerator is at least as large as that of the denominator.

Understanding the nature of these expressions helps in performing operations like integration and solving equations, particularly when using techniques like partial-fraction decomposition.
Degree of Polynomials
The degree of a polynomial is the highest power of the variable in the polynomial. It tells us about the 'size' of the polynomial and is crucial for understanding the behavior of polynomial functions.
When comparing polynomials, degrees help determine which polynomial has leading influence over larger values of the variable.
  • The degree of a numerator and denominator in a rational function can indicate whether you should perform polynomial division before simplifying the expression.
  • If the numerator has a higher degree than the denominator, division should be performed first.

In the context of partial fraction decomposition, understanding degrees is essential. It's especially true because the decomposition requires the numerator to have a degree less than that of the denominator, ensuring a proper fraction form. This concept helps make complex expressions more manageable and easier to integrate or differentiate in calculus.

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Most popular questions from this chapter

$$\text { For } A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], \text { find } A^{2}$$

Ginger talks Gary into putting less money in the money market and more money in the stock (see Exercise 95 ). They place \(10,000\) of their savings into investments. They put some in a money market account earning \(3 \%\) interest, some in a mutual fund that has been averaging \(7 \%\) a year, and some in a stock that rose \(10 \%\) last year. If they put \( 3,000\) more in the stock than in the mutual fund and the mutual fund and stock have the same growth in the next year as they did in the previous year, they will earn \(\$ 840\) in a year. How much money did they put in each of the three investments?

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will eventually reach a maximum height and then fall to the ground. The equation that relates the height \(h\) of a projectile \(t\) seconds after it is projected upward is given by $$h=\frac{1}{2} a t^{2}+v_{0} t+h_{0}$$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\). An object is thrown upward, and the table below depicts the height of the ball \(t\) seconds after the projectile is released. Find the initial height, initial velocity, and acceleration due to gravity. $$\begin{array}{|c|c|} \hline t \text { (seconos) } & \text { HeiGHT (FEET) } \\ \hline 1 & 34 \\ \hline 2 & 36 \\ \hline 3 & 6 \\ \hline \end{array}$$

A company produces three products \(x, y\) and \(z\). Each item of product \(x\) requires 20 units of steel, 2 units of plastic, and 1 unit of glass. Each item of product \(y\) requires 25 units of steel, 5 units of plastic, and no units of glass. Each item of product \(z\) requires 150 units of steel, 10 units of plastic, and 0.5 units of glass. The available amounts of steel, plastic, and glass are \(2400,310,\) and \(28,\) respectively. How many items of each type can the company produce and utilize all the available raw materials?

Determine whether the statements are true or false. $$\text { If } A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], \text { then } A^{-1}=\left[\begin{array}{cc}\frac{1}{a_{11}} & \frac{1}{a_{12}} \\\\\frac{1}{a_{21}} & \frac{1}{a_{22}}\end{array}\right]$$
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