Write each of these rational expressions as a sum of a polynomial and a proper rational expression. I.e., perform polynomial long division. You can check your answer by testing the initial expression and your result at some simple values of the variables to ensure they’re equal.
\(\displaystyle \frac{x^2+1}{x+1} \)
\(\displaystyle \frac{30x+9}{5x+1} \)
\(\displaystyle \frac{x^3+x^2+x+1}{x^3+1} \)
\(\displaystyle \frac{x^3+x^2+x+1}{x^2+x+1} \)
\(\displaystyle \frac{x^3+x^2+x+1}{x+1} \)
\(\displaystyle \frac{x-x^2-x^4}{1-x^3} \)
\(\displaystyle \frac{3x^4-13x^3-8x^2-5x+5}{x-5} \)