Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 10 - Systems of Equations and Inequalities - Section 10.5 Partial Fraction Decomposition - 10.5 Assess Your Understanding - Page 788: 13



Work Step by Step

We can notice from the given rational expression $\dfrac{A(x)}{B(x)}$ that $B(x)$ has only non-repeated linear factors. So, we will write the given rational expression into partial fraction form as: $\displaystyle \frac{4}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1} $ Now, write the RHS with a common denominator. $ \displaystyle \frac{4}{x(x-1)}=\frac{A(x-1)+Bx}{x(x-1}\quad \quad(1)$ Next, equate the numerators to obtain: $Ax-A+Bx=4 \\(Ax+Bx)-A=4 \\(A+B)x-A=4$ Equate the coefficients of the polynomials on the LHS and RHS to obtain: $$-A =4 \implies A=-4$$ Also. $$\\A+B=0 \quad \quad(2)$$ Plug $A=-4$ into the Equation $(2)$ to solve for $B$. $$-4+B=0\implies B=4$$ Thus, the equation $(1)$ becomes: $$ \displaystyle \frac{4}{x(x-1)}=\frac{-4}{x}+\frac{4}{x-1}$$
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