Polynomial Division and Useful Rewrites

Definition A polynomial in $x$ is an expression $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ with real coefficients and whole number exponents. If it is not the zero polynomial, its degree is the largest exponent $n$ with $a_n \ne 0$, and $a_n$ is its leading coefficient. The zero polynomial is given no degree. A nonzero constant has degree $0$.

Definition Let $P$ and $D$ be polynomials with $D$ not the zero polynomial. Then there are unique polynomials $Q$ and $R$ such that

where either $R$ is the zero polynomial or the degree of $R$ is less than the degree of $D$. Here $P$ is the dividend, $D$ is the divisor, $Q$ is the quotient and $R$ is the remainder.

Definition Polynomial long division finds $Q$ and $R$ by repeating one step. Write both polynomials in descending powers, with zero coefficients for missing powers. Divide the leading term of the current dividend (at first $P$ itself, later whatever is left) by the leading term of $D$, add that term to the quotient, multiply it by $D$, and subtract the product from the current dividend, lining up like powers. Stop as soon as what remains is zero or has degree less than the degree of $D$. What remains is $R$.

Definition The remainder $R$ in $P = DQ + R$ is either the zero polynomial or a polynomial of degree less than the degree of $D$. For a linear divisor $x - c$ the remainder is the constant $P(c)$, by setting $x = c$ in the identity. The divisor $D$ is a factor of $P$ exactly when the remainder is zero.

Definition A polynomial identity is an equation between two polynomials that holds for every real input. Two polynomials are identical exactly when their coefficients of each power agree. The division result $P = DQ + R$ is a polynomial identity, so it holds at all real $x$, including the roots of $D$. To check a division, expand $DQ + R$ and compare it with $P$ coefficient by coefficient.

Definition A rational expression $\frac{P(x)}{D(x)}$, with $P$ and $D$ polynomials and $D$ not the zero polynomial, is proper when $P$ is the zero polynomial or the degree of $P$ is less than the degree of $D$. Otherwise it is improper. Dividing both sides of $P = DQ + R$ by $D(x)$ writes an improper expression as a polynomial plus a proper one,

Definition An excluded value of a rational expression $\frac{P(x)}{D(x)}$ is a real number $c$ with $D(c) = 0$. The rewrite $Q(x) + \frac{R(x)}{D(x)}$ equals $\frac{P(x)}{D(x)}$ only for $x$ with $D(x) \ne 0$. This stays true when $R = 0$ and the rewrite looks like a plain polynomial, because the original fraction is still undefined at the roots of $D$.