Final answer:
To find the partial fraction decomposition of the rational function x + 14 / x² - 2x - 8, we need to factor the denominator. The quadratic equation x² - 2x - 8 can be factored as (x - 4)(x + 2). Then, we can set up an equation to find the values of A and B in the partial fraction decomposition. By solving this equation, we find that A = 3 and B = -2. Therefore, the partial fraction decomposition of the rational function is 3 / (x - 4) - 2 / (x + 2).
Step-by-step explanation:
To find the partial fraction decomposition of the rational function - x + 14 / x² - 2x - 8, we first need to factor the denominator. The quadratic equation x² - 2x - 8 can be factored as (x - 4)(x + 2). Therefore, the rational function can be written as (x + 14) / (x - 4)(x + 2).
To find the partial fraction decomposition, we need to find the values of A and B that make the expression:
(x + 14) / (x - 4)(x + 2) = A / (x - 4) + B / (x + 2)
We can do this by multiplying both sides of the equation by (x - 4)(x + 2) and then simplifying. After simplifying, we get the equations:
(x + 14) = A(x + 2) + B(x - 4)
Now we can solve for A and B by substituting appropriate values of x. Let's substitute x = 4:
18 = A(4 + 2) + B(4 - 4)
18 = 6A
A = 3
Now let's substitute x = -2:
12 = A(-2 + 2) + B(-2 - 4)
12 = -6B
B = -2
Therefore, the partial fraction decomposition of the rational function x + 14 / x² - 2x - 8 is: 3 / (x - 4) - 2 / (x + 2).