We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+2
159
2
avatar

Find constants $A$ and $B$ such that \[\frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\] for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair $(A,B)$.

 May 26, 2019
 #1
avatar+4330 
+2

Partial Fraction Decomposition does the job. 

 May 26, 2019
 #2
avatar+23575 
+3

Find constants $A$ and $B$ such that \[\frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\] for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair $(A,B)$.

 

\(\begin{array}{|lrcll|} \hline & \dfrac{x + 7}{x^2 - x - 2} &=& \dfrac{A}{x - 2} + \dfrac{B}{x + 1} \quad | \quad x^2 - x - 2 = (x+1)(x-2) \\\\ & \dfrac{x + 7}{(x+1)(x-2)} &=& \dfrac{A}{x - 2} + \dfrac{B}{x + 1} \quad | \quad \cdot (x+1)(x-2) \\\\ & x + 7 &=& \dfrac{A(x+1)(x-2)}{(x - 2)} + \dfrac{B(x+1)(x-2)}{(x + 1)} \\\\ & \mathbf{x + 7} &=& \mathbf{A(x+1) + B(x-2)} \\ \hline x=2 : & 2 + 7 &=& A(2+1) + B(2-2) \\ & 9 &=& 3 A \\ & A &=& \dfrac{9}{3} \\ & \mathbf{A} &=& \mathbf{3} \\ \hline x=-1 : & -1 + 7 &=& A(-1+1) + B(-1-2) \\ & 6 &=& -3B \\ & B &=& -\dfrac{6}{3} \\ & \mathbf{B} &=& \mathbf{-2} \\ \hline \end{array}\)

 

\(\dfrac{x + 7}{x^2 - x - 2} = \dfrac{3}{x - 2} - \dfrac{2}{x + 1}\)

 

 

laugh

 May 27, 2019

14 Online Users

avatar
avatar