Polynomial Division

Definition When a polynomial $f(x)$ is divided by a polynomial $d(x)$, the polynomial $f$ is the dividend and $d$ is the divisor. The divisor must not be the zero polynomial. Both are written in standard form, with the powers in decreasing order, and every power missing from the dividend gets the coefficient $0$ so that like terms line up during the division.

Definition The division algorithm states that for polynomials $f$ and $d$ with real coefficients, with $d$ not the zero polynomial, there are unique polynomials $q$ and $r$ such that

The polynomial $q$ is the quotient and $r$ is the remainder. The condition on the degree of $r$ is what makes $q$ and $r$ unique.

Definition Long division finds $q$ and $r$ by repeating one step, starting with the whole dividend as the current polynomial. Divide the leading term of the current polynomial by the leading term of $d$ to get the next term of the quotient. Multiply $d$ by that term and subtract the product from the current polynomial. The leading term cancels, so the degree drops with each step. Stop as soon as the current polynomial is $0$ or has degree less than $\deg d$. That last polynomial is the remainder, and the sum of the terms found is the quotient.

Definition Synthetic division is a shortcut for dividing a polynomial $f(x) = a_n x^n + \dots + a_1 x + a_0$ of degree $n \ge 1$ by a divisor of the form $x - c$. List the coefficients $a_n, \dots, a_0$, with zeros for missing powers. The bottom row starts with $b_{n - 1} = a_n$, and each next entry is the next coefficient plus $c$ times the previous bottom entry,

Then the quotient is $q(x) = b_{n - 1} x^{n - 1} + \dots + b_1 x + b_0$ and the remainder is the constant $r$. For a divisor $x + c$ use $-c$, since $x + c = x - (-c)$. For a divisor $ax - b$ with $a \ne 0$, divide by $x - \frac{b}{a}$ to get a quotient $Q(x)$ and a remainder $r$, and then divide $Q(x)$ by $a$. The remainder stays the same, because

Definition The divisor $d$ divides $f$, or is a factor of $f$, when the remainder is $0$, that is when $f(x) = d(x) \, q(x)$ for a polynomial $q$. When the divisor is linear, $x - c$, the remainder is always a constant, because it is $0$ or has degree less than $1$.

Definition Dividing both sides of $f = dq + r$ by $d$ gives

at every $x$ with $d(x) \ne 0$. When $r \ne 0$ the fraction $\frac{r(x)}{d(x)}$ is proper, which means that its numerator has smaller degree than its denominator. The numbers where $d(x) = 0$ stay excluded on both sides.