Question

partial fractions to evaluate:
(x^4+3x^3+2x^2+1)/ (x^2+3x+2)

Answer 1

first do long division since degree of numerator> degree of denominator

Answer 2

x^2
___________________________
x^2+3x+2 | x^4 +3x^3 +2x^2 +1
-(x^4 +3x^3 +2x^2)
--------------------------
1

\[\frac{x^4+3x^3+2x^2+1}{(x+2)(x+1)}=x^2+\frac{1}{(x+2)(x+1)}\]

Answer 3

now you use partial fractions for that fraction

write that \[\frac{1}{(x+2)(x+1)}=\frac{A}{x+2}+\frac{B}{x+1}\]

Answer 4

\[1=A(x+1)+B(x+2)\]
\[1=x(A+B)+(A+2B)\]

Answer 5

=> A+B=0
=>A+2B=1

first equation=> A=-B
pluggin' this into second equation we have
-B+2B=1
B=1

but remember A=-B
so A=-(1)=-1

so we have that
\[\frac{x^4+3x^3+2x^2+1}{x^2+3x+2}=x^2+\frac{-1}{x+2}+\frac{1}{x+1}\]

Answer 6

THANK YOU !