The partial fraction decomposition of 5x2+x+10/(x3+3x2+4x+12) can be written in the form of f(x)/(x+3) and g(x)/(x2+4).Find the values for f(x) and g(x)

The given expression is 5x2+x+10x3+3x2+4x+12 It can be rewritten as…

Answered: The partial fraction decomposition of…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section: Chapter Questions

Problem 7CT

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The partial fraction decomposition of 5x²+x+10/(x³+3x²+4x+12) can be written in the form of f(x)/(x+3) and g(x)/(x²+4).

Find the values for f(x) and g(x)

Expert Solution

Step 1

The given expression is

$\frac{5 x^{2} + x + 10}{x^{3} + 3 x^{2} + 4 x + 12}$

It can be rewritten as

$\begin{array}{rcl} \frac{5 x^{2} + x + 10}{x^{3} + 3 x^{2} + 4 x + 12} & = & \frac{5 x^{2} + x + 10}{x^{2} (x + 3) + 4 (x + 3)} \\ = & \frac{5 x^{2} + x + 10}{(x + 3) (x^{2} + 4)} \end{array}$

The form of partial decomposition of given expression is

$\frac{5 x^{2} + x + 10}{(x + 3) (x^{2} + 4)} = \frac{A}{x + 3} + \frac{B x + C}{x^{2} + 4}$

Step 2

Simplify the above expression

$\frac{5 x^{2} + x + 10}{(x + 3) (x^{2} + 4)} = \frac{A (x^{2} + 4) + (x + 3) (B x + C)}{(x + 3) (x^{2} + 4)}$

Since, the denominator are same on both sides. So,comparing numerator on both sides.

Therefore,

$5 x^{2} + x + 10 = A (x^{2} + 4) + (B x + C) (x + 3) 5 x^{2} + x + 10 = A x^{2} + 4 A + B x^{2} + 3 x B + C x + 3 C$

Collect up like terms on right side

$= (A + B) x^{2} + x (C + 3 B) + 4 A + 3 C$

Now, comparing the coefficients on both sides

$A + B = 5 . . . (1) 3 B + C = 1 . . . (2) 4 A + 3 C = 10 . . . (3)$

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