Polynomial division with the algorithm

The first term of the quotient is the ratio of the highest-degree terms of the dividend and the divisor, that is, $\frac{x^3}{x}=x^2$; the divisor is multiplied by $x^2$ and the product is subtracted from the dividend, and the partial remainder is $x^2-x+3$. The partial remainder has degree $2$, greater than the degree of the divisor, so the algorithm continues: the next term is $\frac{x^2}{x}=x$ and the new partial remainder is $-2x+3$. This one also has degree equal to that of the divisor, therefore a last step is carried out with the term $\frac{-2x}{x}=-2$, which leaves the remainder $5$. The remainder has degree zero, less than the degree of the divisor, so the algorithm stops.

The check uses the identity principle: $(x+1)(x^2+x-2)+5=x^3+2x^2-x-2+5=x^3+2x^2-x+3$, which is the dividend.

The dividend lacks the term of degree $2$, so it is rewritten with the coefficient $0$ in place of the missing degree; in this way every column of the algorithm stays aligned.

The first term of the quotient is $\frac{2x^4}{x^2}=2x^2$ and the first subtraction leaves $-3x^3+4x^2+x-1$. Since the degree of the partial remainder is $3$, greater than the degree of the divisor, the algorithm continues with $\frac{-3x^3}{x^2}=-3x$, which leaves $4x^2-5x-1$. The degree is still equal to that of the divisor, therefore the last term of the quotient is $\frac{4x^2}{x^2}=4$ and the remainder is $-5x+7$. The remainder has degree $1$, less than $2$, so the division is complete: when the divisor has degree $2$, the remainder can contain both a term in $x$ and a constant term.

The three steps of the algorithm produce the terms $x^2$, $-x$ and $1$ of the quotient; after the last subtraction the remainder is $0$, so the division is exact. By Ruffini’s theorem this means that $2$ is a root of the dividend, and indeed $P(2)=8-12+6-2=0$. The divisor $x-2$ is therefore a factor of the dividend, and the division provides the factorization $P(x)=(x-2)(x^2-x+1)$.