Synthetic division, the Remainder Theorem and the Factor Theorem

A shortcut for one common case

Long division always works, but it's slow. When the divisor has the form \(x - c\), a linear binomial whose leading coefficient is 1, there's a much faster method: synthetic division. It keeps only the numbers and drops the variables.

Divisors like \(x - 4\), \(x + 1\) (which is \(x - (-1)\)) or \(x - \frac{2}{3}\) all qualify. The number \(c\) is the value that makes the divisor zero: for \(x + 1\), \(c = -1\).

How it works

Let's redo a division from the previous lesson, \((3x^3 - 5x^2 + 4x - 7) \div (x - 2)\), where \(c = 2\).

The Remainder Theorem

There's an even faster way to get the remainder: don't divide at all.

Why? By the definition of division, \(P(x) = (x - c)\,Q(x) + R\). Plug in \(x = c\): the first part becomes \(0 \cdot Q(c) = 0\), which leaves \(P(c) = R\).

The Factor Theorem

The Remainder Theorem has an immediate consequence, and it's one of the most useful facts in algebra.

A number \(c\) with \(P(c) = 0\) is called a zero (or root) of the polynomial. Finding a zero hands you a linear factor, which is exactly what you need to factor higher-degree polynomials.

You don't have to guess zeros at random.

Practice