function ( partial fraction)
posted by kingwendu on .
if f(x) ax^2+bx+c , g(x) :1/3x^2+2 and fg(x)= 18x^4+24x^2+11/(3x^2+2)^2. find a ,b and c

Well, thinking fg(x) means f(x) * g(x) I get no consistent solution. So, making
fg(x) = f(g(x)) we have
f(g) = ag^2 + bg + c
= a/(3x^2+2)^2 + b/(3x^2+2) + c
= [a + b(3x^2+2) + c(3x^2+2)^2] / (3x^2+2)^2
= 9cx^4 + (12c+3b)x^2 + (4c+2b+a)
= 18x^4 + 24x^2 + 11
That makes c=2, b=0, a=3, so f(x) = 2x^2 + 3