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1

Solved example of algebraic fractions

$\frac{x^3+x+5}{x+1}$

2

Let's divide the polynomial by $x+1$ using synthetic division (also known as Ruffini's rule). First, write all the coefficients of the polynomial in the numerator in descending order based on grade (putting a zero if a term doesn't exist). Then, take the first coefficient ($1$) and multiply it by the root of the denominator ($-1$). Add the result to the second coefficient and multiply this by $-1$ and so on

$\left|\begin{matrix}1 & 0 & 1 & 5 \\ & -1 & 1 & -2 \\ 1 & -1 & 2 & 3\end{matrix}\right|-1$

3

In the last row appear the new coefficients of the polynomial. Use these coefficients to rewrite the new polynomial with a lower grade, and the remainder ($3$) divided by the divisor

$x^{2}-x+2+3x^{-1}+\frac{3}{x+1}$