How do you write the partial fraction decomposition of the rational expression # 3/(x ^(2) + 16x +5)#?

2 Answers
Dec 14, 2015

#3/(x^2+16x+5)#

Explanation:

Notice how the denominator cannot be factored (within the real numbers).

This fraction is already as simple as it can get.

Dec 14, 2015

#3/(x^2+16x+5) = 3/(2 sqrt(59)(x+8-sqrt(59)))-3/(2 sqrt(59)(x+8+sqrt(59)))#

Explanation:

#x^2+16x+5 = x^2+16x+64-59#

#=(x+8)^2-(sqrt(59))^2#

#=(x+8-sqrt(59))(x+8+sqrt(59))#

So to find the partial fraction decomposition, solve:

#3/((x+8-sqrt(59))(x+8+sqrt(59)))#

#= A/(x+8-sqrt(59)) + B/(x+8+sqrt(59))#

#= (A(x+8+sqrt(59))+B(x+8-sqrt(59)))/((x+8-sqrt(59))(x+8+sqrt(59)))#

#=((A+B)x + 8(A+B) + (A-B)sqrt(59))/((x+8-sqrt(59))(x+8+sqrt(59)))#

Equating the coefficient of #x#, we find #A+B = 0#. So #B = -A#.

Equating the remaining constant term we find #2 A sqrt(59) = 3#

Hence #A=3/(2 sqrt(59))# and:

#3/(x^2+16x+5) = 3/(2 sqrt(59)(x+8-sqrt(59)))-3/(2 sqrt(59)(x+8+sqrt(59)))#