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Chapter 12: Problem 8

Write the partial fraction decomposition for the expression. $$ \frac{3 x^{2}-x+1}{x(x+1)^{2}} $$

Short Answer

Expert verified
The partial fraction decomposition of the expression \(\frac{3x^{2}-x+1}{x (x+1)^{2}}\) is \(\frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\), where values of A, B, and C are found by solving the polynomial equation created in step 4.

Step by step solution

01

Identify the denominator's factors

You must first factorize the denominator thoroughly. In this case, the expression \((x(x+1)^{2})\) has already been factorized and has two unique factors: \(x\) and \((x+1)^2\).
02

Set up the partial fractions

Partial fractions are set up based on the factors of the denominator. The aim is to write the original fraction as sum of these fractions. You must bear in mind that the numerator of each fraction is always one degree less than the denominator. In this case, the factors are \(x\) with degree 1 and \((x + 1)^{2}\) with degree 2. Hence, the partial fractions set up would be \(\frac{A}{x}\) and \(\frac{Bx + C}{(x+1)^{2}}\). So, \(\frac{3x^{2}-x+1}{x(x+1)^{2}} = \frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\).
03

Clear out the fractions

To get rid of the denominators in the equation, you multiply both sides of the equation by the original denominator: \(x(x+1)^{2}\). Doing this, you get: \(3x^{2}-x+1 = Ax(x+1)^{2} + Bx(x) + C(x)\).
04

Solve for A, B, and C

Collect terms to write the equation as a polynomial: \(Ax^{3} + 2Ax^{2} + Ax + Bx^2 + Cx = 3x^{2} - x + 1\). Here, to get the values of A,B,C you could equate coefficients of powers of x, or substitute suitable values of x to get system of linear equations and solve those equations. Choosing convenient values of x can solve the equation system.
05

Substitute values of A, B, C into partial fractions

Once you have the values of A, B, and C from Step 4, substitute those values back in to the equations for the partial fractions found in Step 2: \(\frac{3x^{2}-x+1}{x(x+1)^{2}}\) = \(\frac{A}{x} + \frac{Bx + C}{(x+1)^{2}}\) and simplify. This will result in the final answer.

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Most popular questions from this chapter

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} \frac{1}{x+1} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x, n=4 $$

Probability The probability of finding between \(a\) and \(b\) percent iron in ore samples is modeled by \(P(a \leq x \leq b)=\int_{a}^{b} 2 x^{3} e^{x^{2}} d x, \quad 0 \leq a \leq b \leq 1\) (see figure). Find the probabilities that a sample will contain between (a) \(0 \%\) and \(25 \%\) and (b) \(50 \%\) and \(100 \%\) iron.

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1) $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$
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