Question #ff941

1 Answer
Dec 13, 2017

#(x^3+x^2+7)/(x^2+9)^2=-(1+sqrt(3))/(3(x^2+9))+(10+3sqrt(3))/(x^2+9)^2=(-1+sqrt(3))/(3(x^2+9))+(10-3sqrt(3))/(x^2+9)^2#

Explanation:

#(x^3+x^2+7)/(x^2+9)^2=A/(x^2+9)+B/(x^2+9)^2#

#=(A(x^2+9))/((x^2+9)^2)+B/((x^2+9)^2)#

#=(A(x^2+9)+B)/((x^2+9)^2)#

#x^3+x^2+7=A(x^2+9)+B#

Using #x=sqrt(3)#

#sqrt(3)^3+sqrt(3)^2+7=A(sqrt(3)^2+9)+B#

#B=10+3sqrt(3)#

#x^3+x^2+7=A(x^2+9)+10+3sqrt(3)#

With #x=0#

#7=9A+10+3sqrt(3)#

#A=(7-10-3sqrt(3))/9=-(1+sqrt(3))/3#

#(x^3+x^2+7)/(x^2+9)^2=(-(1+sqrt(3))/3)/(x^2+9)+(10+3sqrt(3))/(x^2+9)^2=-(1+sqrt(3))/(3(x^2+9))+(10+3sqrt(3))/(x^2+9)^2#

Using #x=-sqrt(3)#

#(-sqrt(3))^3+(-sqrt(3))^2+7=A((-sqrt(3))^2+9)+B#

#B=10-3sqrt(3)#

#x^3+x^2+7=A(x^2+9)+10-3sqrt(3)#

With #x=0#

#7=9A+10-3sqrt(3)#

#A=(7-10+3sqrt(3))/9=(-1+sqrt(3))/3#

#(x^3+x^2+7)/(x^2+9)^2=((-1+sqrt(3))/3)/(x^2+9)+(10-3sqrt(3))/(x^2+9)^2=(-1+sqrt(3))/(3(x^2+9))+(10-3sqrt(3))/(x^2+9)^2#