The partial fraction decomposition is [tex]\frac{8x + 19}{(x + 8)(x - 1)} = \frac{5}{x + 8} + \frac{3}{x - 1}[/tex]
The fraction is given as:
[tex]\frac{8x + 19}{(x + 8)(x - 1)}[/tex]
Split the fraction as follows:
[tex]\frac{8x + 19}{(x + 8)(x - 1)} = \frac{A}{x + 8} + \frac{B}{x - 1}[/tex]
Take the LCM
[tex]\frac{8x + 19}{(x + 8)(x - 1)} = \frac{Ax -A + Bx + 8B}{(x + 8)(x -1)}[/tex]
Cancel the common factors
8x + 19 = Ax - A + Bx + 8B
By comparison, we have:
Ax + Bx = 8x
-A + 8B = 19
This gives
A + B = 8
-A + 8B = 19
Add both equations
9B = 27
Divide by 9
B = 3
Substitute B = 3 in A + B = 8
A + 3 = 8
Solve for A
A = 5
So, we have:
[tex]\frac{8x + 19}{(x + 8)(x - 1)} = \frac{5}{x + 8} + \frac{3}{x - 1}[/tex]
Hence, the partial fraction decomposition is [tex]\frac{8x + 19}{(x + 8)(x - 1)} = \frac{5}{x + 8} + \frac{3}{x - 1}[/tex]
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