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Chapter 8: Problem 18

Give the partial fraction decomposition for the following expressions. $$\frac{11 x-10}{x^{2}-x}$$

Short Answer

Expert verified
Answer: The partial fraction decomposition is $$\frac{11x - 10}{x^2 - x} = \frac{10}{x} + \frac{1}{x-1}$$.

Step by step solution

01

Factor the denominator

First, we need to factor the denominator of the given expression. The denominator is given as $$x^2-x$$. Take the common factor, x, out of both terms to get: $$x^2-x=x(x-1)$$
02

Write the general partial fraction decomposition

Now, we can write the general partial fraction decomposition which consists of a sum of partial fractions. Each partial fraction will have one of the factors we found in step 1 as its denominator and an unknown constant in the numerator: $$\frac{11x - 10}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}$$ Note that A and B are the coefficients we need to find.
03

Expand and equate the numerators

Next, we need to clear the fractions. Multiply both sides by the common denominator, $$x(x-1)$$: $$(x(x-1))\left(\frac{11x - 10}{x(x-1)}\right) = (x(x-1))\left(\frac{A}{x} + \frac{B}{x-1}\right)$$ Simplifying, we get: $$11x-10 = A(x-1) + Bx$$
04

Solve for the coefficients

Now we need to solve for the unknown coefficients A and B. We can do this by comparing the coefficients of x for the left-hand side and right-hand side: Comparing coefficients of $$x$$: $$11 = A + B$$ Comparing constant terms: $$-10 = -A$$ The second equation gives us directly: $$A = 10$$ Substituting this value in the first equation, we get: $$11 = 10 + B$$ $$B = 1$$
05

Rewrite the original expression

Using the coefficients we found in step 4, we can now rewrite the original expression as a sum of partial fractions: $$\frac{11x - 10}{x(x-1)} = \frac{10}{x} + \frac{1}{x-1}$$ This is the partial fraction decomposition of the given expression.

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Key Concepts

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

Factorization
Factorization is a crucial step in simplifying expressions, particularly when dealing with partial fraction decompositions. This process involves breaking down a complex expression into simpler, smaller parts, which are easier to manage and solve. In the context of our exercise, we began by factoring the denominator. We have the expression \( x^2 - x \). The first task is to look for common factors.

  • Here, both terms of the denominator, \( x^2 \) and \( -x \), share a common factor of \( x \).
  • When we factor \( x \) out from both terms, the expression becomes \( x(x-1) \).
  • This is significant because it allows us to break down the rational expression into partial fractions, each with these factors as denominators.
By recognizing that \( x \) is common and extracting it, we open the door to handling each part of the expression separately. Factorization may seem simple but it's a powerful tool in algebra that helps in reducing the complexity of problems.
Coefficients
Coefficients play a significant role when you perform partial fraction decomposition. They represent unknown values that must be determined to express the rational expression appropriately. In the given exercise, after factorizing the denominator, we break the expression into two separate fractions:

  • \( \frac{A}{x} \)
  • \( \frac{B}{x-1} \)
Here, \( A \) and \( B \) are the coefficients we need to find. Setting up the equation:\[11x - 10 = A(x-1) + Bx\]The steps involve comparing coefficients from each side of the equation:
  • From the \( x \) terms, \( A + B = 11 \).
  • From the constant terms, \( -A = -10 \) which gives \( A = 10 \).
Replacing \( A \) in the first equation, we solve for \( B \):\[B = 11 - 10 = 1\]Understanding how to derive these coefficients ensures the completeness and correctness of the partial fraction decomposition.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. They can often appear complex, but with techniques like partial fraction decomposition, they become more manageable. In our exercise, the rational expression is initially given as:\[\frac{11x - 10}{x^2 - x}\]The task of partial fraction decomposition is to express this as:\[\frac{A}{x} + \frac{B}{x-1}\]Once we determine the necessary coefficients, \( A \) and \( B \), we can decompose the original rational expression into simpler parts. This decomposition helps in several ways:
  • It simplifies the process of integration in calculus, as integrating separate fractions is often easier.
  • It allows for easy evaluation of limits and provides insights into the behavior of the expression at different values.
Mastering the manipulation and simplification of rational expressions is essential for a deeper understanding of algebra and calculus topics.

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Most popular questions from this chapter

The nucleus of an atom is positively charged because it consists of positively charged protons and uncharged neutrons. To bring a free proton toward a nucleus, a repulsive force \(F(r)=k q Q / r^{2}\) must be overcome, where \(q=1.6 \times 10^{-19} \mathrm{C}(\) coulombs ) is the charge on the proton, \(k=9 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}, Q\) is the charge on the nucleus, and \(r\) is the distance between the center of the nucleus and the proton. Find the work required to bring a free proton (assumed to be a point mass) from a large distance \((r \rightarrow \infty)\) to the edge of a nucleus that has a charge \(Q=50 q\) and a radius of \(6 \times 10^{-11} \mathrm{m}\).

Determine whether the following integrals converge or diverge. $$\int_{0}^{\infty} \frac{d x}{e^{x}+x+1}$$

Another form of \(\int \sec x \, d x\) a. Verify the identity sec \(x=\frac{\cos x}{1-\sin ^{2} x}\) b. Use the identity in part (a) to verify that \(\int \sec x \, d x=\frac{1}{2} \ln \left|\frac{1+\sin x}{1-\sin x}\right|+C\)

It can be shown that $$\begin{array}{l}\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x= \\\\\quad\left\\{\begin{array}{ll}\frac{1 \cdot 3 \cdot 5 \cdot \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2} & \text { if } n \geq 2 \text { is an eveninteger } \\\\\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n} & \text { if } n \geq 3 \text { is an odd integer. }\end{array}\right.\end{array}$$ a. Use a computer algebra system to confirm this result for \(n=2,3,4,\) and 5 b. Evaluate the integrals with \(n=10\) and confirm the result. c. Using graphing and/or symbolic computation, determine whether the values of the integrals increase or decrease as \(n\) increases.

Let \(R\) be the region between the curves \(y=e^{-c x}\) and \(y=-e^{-c x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c>0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves; see the Guided Project The exponential Eiffel Tower. ) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R .\) Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\)
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