Question

,
$\frac{ {x}^{2} + x + 1}{ {x}^{3} + {x}^{2} - 6x}$
make it partial fraction ​

Answer 1

x²+x+1/ x³+x²-6x

denominator:

x(x²+x-6) = x(x²+3x-2x-6) =x(x+3)(x-2)

x²+x+1/ x³+x²-6x = a/x + b/x+3 + c/x-2

=a(x²+x-6) +b(x²-2x)+c(x²+3x) / x³+x²-6x

a+b+c = 1 ------A

a-2b+3c = 1 ------B

-6a = 1 ------C

a = -1/6

substitute in A

b+c =1+1/6 = 7/6 ----C

substitute in B

-2b+3c = 7/6 -----D

b+c =-2b+3c

3b = 2c

c=3/2b

substitute in C

b+3/2b = 7/6

5b =7/3

b = 7/15

c = 3/2×7/15 =7/10

so partial fraction is

-1/6x + 7/15(x+3) + 7/10(x-2)