Rational Zeros Theorem and Factoring

Definition A rational number is a number of the form $\frac{p}{q}$ with integers $p$ and $q$ and $q \ne 0$. The fraction is in lowest terms, or reduced, when $q > 0$ and $p$ and $q$ have no common positive divisor other than $1$. Every rational number has exactly one such reduced form. An integer $m$ has the reduced form $\frac{m}{1}$.

Definition Let

have integer coefficients, with $n \ge 1$, $a_n \ne 0$ and $a_0 \ne 0$. If the reduced fraction $\frac{p}{q}$ is a zero of $P$, then $p$ divides $a_0$ and $q$ divides $a_n$.

The reason fits in a few lines. Multiplying $P(p/q) = 0$ by $q^n$ gives

Every term except the last contains the factor $p$, so $p$ divides $a_0 q^n$. Since $p$ and $q$ have no common prime factor, neither do $p$ and $q^n$, so $p$ divides $a_0$. In the same way every term except the first contains $q$, so $q$ divides $a_n p^n$ and therefore $a_n$.

Definition For integers $d \ne 0$ and $m$, the number $d$ divides $m$ when $m = dk$ for some integer $k$. For the theorem you need all divisors of the constant term $a_0$ and of the leading coefficient $a_n$. A nonzero integer $m$ has finitely many divisors, and they come in pairs $\pm d$ where $d$ runs through the positive divisors of $\lvert m \rvert$. Two cases need a step first.

1. If $a_0 = 0$, then $x$ is a factor. Write $P(x) = x^k R(x)$ with $R(0) \ne 0$ and apply the theorem to $R$.
2. If every coefficient is rational but some are not integers, multiply $P$ by a common denominator of the coefficients. This gives a polynomial with integer coefficients and the same zeros.

Definition Under the hypotheses of the theorem, the rational root candidates of $P$ are all numbers

with repeated values listed once. Every rational zero of $P$ is on this list. The list is finite.

Definition A candidate $r$ is a zero of $P$ exactly when $P(r) = 0$. By the Remainder Theorem this is the same as the remainder of $P(x)$ divided by $x - r$ being $0$, so either direct evaluation or synthetic division decides it. Two values are quick to get. The value $P(1)$ is the sum of the coefficients, and $P(-1)$ is the same sum after switching the sign of each coefficient of an odd power of $x$.

Definition Synthetic division divides $P(x)$ by $x - r$ using the coefficients alone, with $0$ for each missing power. Bring down the leading coefficient, then repeatedly multiply the newest bottom entry by $r$ and add the product to the next coefficient. The last bottom entry is the remainder $P(r)$, and the entries before it are the coefficients of the quotient. When $r$ is a zero the remainder is $0$, so

where the quotient $Q$, called the depressed polynomial, has degree one less than $P$. The remaining zeros of $P$ are exactly the zeros of $Q$, so the search continues on $Q$. Every rational zero of $Q$ is also a rational zero of $P$, so it is already on the list for $P$, and candidates that failed for $P$ can be skipped. A zero can repeat, so test $r$ itself again on $Q$. When $Q$ reaches degree $2$, finish with factoring or the quadratic formula.

Definition The Rational Zeros Theorem says only that every rational zero is on the candidate list. It does not say that any candidate is a zero, and it says nothing about zeros that are irrational or not real. A polynomial with integer coefficients may have no rational zeros at all, even when it has real zeros.