Question

Set up and solve an appropriate system of linear equations to answer the question. The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of mathematics; for example, it arises in calculus when we need to integrate a rational function and in discrete mathematics when we use generating functions to solve recurrence relations. The decomposition of a rational function as a sum of partial fractions leads to a system of linear equations. In the problem, find the partial fraction decomposition of the given form. (The capital letters denote constants.) $$ \frac{x^2 - 5x + 5}{x^3 + 2x^2 + x} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} $$ $$(A, B, C) = (\boxed{\phantom{123456789}})$$

          Set up and solve an appropriate system of linear equations to answer the question.
The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of mathematics; for example, it arises in calculus when we need to integrate a rational function and in discrete mathematics when we use generating functions to solve recurrence relations. The decomposition of a rational function as a sum of partial fractions leads to a system of linear equations. In the problem, find the partial fraction decomposition of the given form. (The capital letters denote constants.)
$$ \frac{x^2 - 5x + 5}{x^3 + 2x^2 + x} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} $$
$$(A, B, C) = (\boxed{\phantom{123456789}})$$

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set up and solve an appropriate system of linear equations to answer the question the process of adding rational functions ratios of polynomials by placing them over a common denominator is 43661

Added by Bradley A.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Ashley Volpe

06/17/2020

Step 1

The given form is: $$ \frac{x^2 - 5x + 5}{x^3 + 2x^2 + x} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} $$ First, factor the denominator of the left side: $$ x^3 + 2x^2 + x = x(x^2 + 2x + 1) = x(x+1)^2 $$ So the equation becomes: $$ \frac{x^2 - 5x + Show more…

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Set up and solve an appropriate system of linear equations to answer the question. The process of adding rational functions (ratios of polynomials) by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of mathematics; for example, it arises in calculus when we need to integrate a rational function and in discrete mathematics when we use generating functions to solve recurrence relations. The decomposition of a rational function as a sum of partial fractions leads to a system of linear equations. In the problem, find the partial fraction decomposition of the given form. (The capital letters denote constants.) $$ \frac{x^2 - 5x + 5}{x^3 + 2x^2 + x} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} $$ $$(A, B, C) = (\boxed{\phantom{123456789}})$$

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Transcript

00:01 Alright, so we're going to start this with some multiplication.

00:05 We notice that we have this polynomial, and it's equal to a times something plus b time something plus c time something.

00:14 So we're going to start with doing that multiplication.

00:17 We have a times x squared plus 5x plus 6, which we know is a x squared plus 5 a x plus 6 a.

00:30 The second one is the b one, and that's going to be b times x squared plus 4x plus 3.

00:39 And we know that that is bx squared plus 4bx plus 3b.

00:47 And then the final one we have is c times x squared plus 3x plus 2.

00:55 And we know that that is cx squared plus 3cx plus 2c.

01:03 Now the beginning of this problem told us that we had x squared plus x plus 1.

01:09 And so we know that this column should equal our x squared.

01:13 This one should be our x and this should be our 1.

01:17 And so we basically have built our three equations right here.

01:23 Now if we just use the numbers and the numbers, letters, we have our three equations.

01:30 So one of my equations is a plus b.

01:33 Those are the coefficients of x squared and it should equal the coefficient of the x squared in the answer.

01:41 We also have 5a plus 4b plus 3c equals 1 and we have 6a plus 3b plus 2c plus 3b plus 2c.

01:57 And we have 6a plus 3b plus 2c, equals 1.

02:04 We have a system.

02:06 We need to solve that system.

02:08 So we're going to do this in two parts.

02:09 We're going to do the first and second together.

02:12 And then we're going to do the first and third together and eliminate our c terms...

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