- #1
chwala
Gold Member
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- Homework Statement
- Find the values of ##A,B## and ##C## if the expression
##Ax(x-2)(x+3)+Bx(x-2)+Cx(x+3)+ (x-2)(x+3)##
has a constant value for all values of ##x##.
- Relevant Equations
- Partial fractions
My working follows here,
Let
##(x-2)[Ax(x+3)+Bx] + (x+3)[Cx +(x-2)]=0##
##(x-2)(x+3) \left[Ax+\dfrac{Bx}{(x+3)}\right]+(x+3)(x-2)\left[\dfrac{Cx}{(x-2)} +1\right]##=0
##⇒Ax+\dfrac{Bx}{(x+3)} = -\left[\dfrac{Cx}{(x-2)} +1\right]##
##Ax(x+3)(x-2)+Bx(x-2) = -[Cx(x+3)+(x-2)(x+3)]##
On comparing coefficients we end up with;
##A=0##
...and the simultaneous equation;
##B+C=-1##
##-2B+3C=-1##
Giving me
##B=-\dfrac{2}{5}## and ##C=-\dfrac{3}{5}##
There could be a better approach...And also your insight on my steps are welcome. Cheers!
Let
##(x-2)[Ax(x+3)+Bx] + (x+3)[Cx +(x-2)]=0##
##(x-2)(x+3) \left[Ax+\dfrac{Bx}{(x+3)}\right]+(x+3)(x-2)\left[\dfrac{Cx}{(x-2)} +1\right]##=0
##⇒Ax+\dfrac{Bx}{(x+3)} = -\left[\dfrac{Cx}{(x-2)} +1\right]##
##Ax(x+3)(x-2)+Bx(x-2) = -[Cx(x+3)+(x-2)(x+3)]##
On comparing coefficients we end up with;
##A=0##
...and the simultaneous equation;
##B+C=-1##
##-2B+3C=-1##
Giving me
##B=-\dfrac{2}{5}## and ##C=-\dfrac{3}{5}##
There could be a better approach...And also your insight on my steps are welcome. Cheers!
Last edited: