Hello there. To solve this question, we'll have to remember some properties about dividing polynomials.
Given the polynomials, we want to evaluate the division:
[tex](1+3x+x^4)\div(3-2x+x^2)[/tex]
Rewriting it the way we perform long division:
We start with the higher degree terms, namely x^4 and x².
Dividing x^4 by x², we get x². Now we multiply every term from the division by this factor and subtract from the term being divided.
Now, we have a 2x³ as the higher degree term from the term being divided. Dividing it by x², we get 2x. Multiply each term of the divisor and subtract from it.
Finally, the highest degree term from the term being divided is x². Dividing it by x², we get 1. Multiply each term of the divisor and subtract it from the dividend.
Now, the highest degree term from the dividend is -x, when the highest degree term from the divisor is x². We cannot proceed with the long division anymore.
It means that we have a quotient:
[tex]x^2+2x+1[/tex]
And a remainder:
[tex]-x-2[/tex]
Notice if we rewrite it as:
[tex]x^4+3x^2+1=(x^2-2x+3)\cdot(x^2+2x+1)-x-2[/tex]
We have the division P(x)/D(x) written in the form:
[tex]P(x)=D(x)\cdot Q(x)+R(x)[/tex]
Where Q(x) and R(x) are the quotient and remainder polynomials.
