The first thing we should do when trying to divide two rational functions is to see if the denominator is a zero of the numerator.
Let #f(x)=2x^4-5x^3-8x^2+17x+1#
#f(sqrt2)=-7+7sqrt2#
We know that any rational function can be written as: #f(x)=p(x)q(x)+r(x)#. Thus we can say:
#2 x^4 - 5 x^3 - 8 x^2 + 17 x + 1 =(ax^2+bx+c)(x^2-2)+7x-7#
This is because when #(x^2-2)=0#, the remainder was #(7x-7)#. So #r(x)=7x-7#. We already know #q(x)# and we need to work out #p(x)#.
#2x^4-5x^3-8x^2+10x+8=(ax^2+bx+c)(x^2-2)#
#ax^4=2x^4rArra=2#
#bx^3=-5x^3rArrb=-5#
#-2c=8rArrc=-4#
#2x^4-5x^3-8x^2+10x+8=(2x^2-5x-4)(x^2-2)#
#2 x^4 - 5 x^3 - 8 x^2 + 17 x + 1=(2x^2-5x-4)(x^2-2) +7x-7#
#(2 x^4 - 5 x^3 - 8 x^2 + 17 x + 1 )/(x^2 - 2)= (2 x^2 - 5 x - 4) + (7 x - 7
)/(x^2-2)#
