xample 1: Write the partial fraction decomposition of the following expression.(20x + 35)/(x + 4)2

The given expression is (20x + 35)/((x + 4)^2).

The first step in partial fraction decomposition is to express the given rational function as the sum of simpler fractions.

Since the denominator is a repeated linear factor (x + 4)^2, we can express the given function as:

(20x + 35)/((x + 4)^2) = A/(x + 4) + B/(x + 4)^2

Next, we clear the fractions by multiplying through by the common denominator, (x + 4)^2:

20x + 35 = A*(x + 4) + B

Now, we can solve for A and B by substituting suitable values for x.

Let's choose x = -4, because this will simplify the equation significantly:

20*(-4) + 35 = A*(-4 + 4) + B -45 = B

Next, we can substitute x = 0 into the equation to solve for A:

200 + 35 = A(0 + 4) - 45 35 = 4A - 45 80 = 4A A = 20

So, the partial fraction decomposition of the given expression is:

(20x + 35)/((x + 4)^2) = 20/(x + 4) - 45/(x + 4)^2