Factor of 1+x3=(1+x)(1−x+x2)
∴ x1+x3=x(1+x)(1−x+x2)
Let x(1+x)(1−x+x2)=A1+x+Bx+C1−x+x2....(A)
x=A(1−x+x2)+(Bx+C)(1+x)
x=A−Ax+Ax2+Bx+Bx2+C+Cx
Comparing the coefficients of x,x2 and constant terms both sides
A+B=0......(1)
−A+B+C=1........(2)
A+C=0........(3)
equation (1)-(3)
A+B=0.......(1)
A+C=0......(3)
-------------------------
B−C=0.......(4)
equation (2)-(4)
−A+B+C=1
B−C=0
---------------------
−A+2B=1.......(5)
equation (1)+equation (5)
A+B=0......(1)
−A+2B=1......(5)
------------------------------
3B=1
B=13
by equation (1) A+B=0⇒A=−13
by equation (3) A+C=0⇒C=13
x1+x3=−13(1−x)+13x+131−x+x2
=13(1−x)+x+13(1−x+x2)