When the degree of the numerator is the same or greater than
the degree of the denominator the expression must first be
divided out by long division:
4
x²+4x+4)4x²+11x+ 0
4x²+16x+16
-5x-16
= 4 +
Next we find the partial fraction
decomposition of and when we finish
we will add it to the 4:
Factor the denominator as (x+2)(x+2) or (x+2)².
That is a power in the denominator, so we must include in the
decomposition factors with denominators of it and all lower
powers, so we assume A and B such that:
= +
Clear of fractions by multiplying through by
the LCD (x+2)²:
-5x - 16 = A + B(x + 2)
-5x - 16 = A + Bx + 2B
Equate the coefficients of x
-5 = B
Equate the constants:
-16 = A + 2B
Substitute -5 for B
-16 = A + 2(-5)
-16 = A - 10
-6 = A
So
= +
And therefore the original:
= 4 + = 4 + +
or a bit simpler:
4 - -
Edwin
