When we divide polynomials, remember that the rules of the signs are inverted, that means that if we have (+)(+) = - and (-)(-)=- and so on. Then, we have the following:
first we divide 4x^4 by x^2 (since it's the main term of the divisor) , we get:
[tex]\frac{4x^4}{x^2}=4x^2[/tex]
Then we write 4x^2 on our division box, and proceed like a normal long division but with the law of signs changed, to get the first substraction:
next, we do the same but now with the term 6^3, to get the following:
finally, we have the main term 17x^2, since it has the same exponent as our main term in the divisior, this is the last step to find the residue:
we have that the residue is 35x+25, then, the expression 4x^2-2x^3+x^2-5x+8 can be written as:
[tex]4x^4-2x^3+x^2-5x+8=(x^2-2x-1)(4x^2+6x+17)+\frac{35x+25}{x^2-2x-1}[/tex]
therefore, the result of dividing 4x^4-2x^3+x^2-5x+8 by x^2-2x-1 is:
[tex]\begin{gathered} \text{ Quotient:} \\ 4x^2+6x+17\text{ } \\ \text{ Residue:} \\ 35x+25 \end{gathered}[/tex]
