How Computers Evaluate Polynomials: Synthetic Division

Dividing a polynomial by x − c with synthetic division takes a few multiply-and-add steps, and the remainder is exactly P(c). That's Horner's method, the way calculators and computers evaluate polynomials. When the remainder is zero, x − c is a factor.

How it works

Dividing polynomials, the remainder theorem and the factor theorem

In one sentence

For a polynomial $P$ and a number $c$ , division by $x - c$ gives a unique quotient $Q$ , one degree lower, and a constant remainder $R$ with $P(x) = (x - c)Q(x) + R$ . Remainder theorem: $R = P(c)$ . Factor theorem: $x - c$ is a factor of $P$ exactly when $P(c) = 0$ .

General formula

{\htmlData{sym=0}{P(x)}} = (x - {\htmlData{sym=1}{c}})\,{\htmlData{sym=2}{Q(x)}} + {\htmlData{sym=3}{R}}, \qquad {\htmlData{sym=3}{R}} = P({\htmlData{sym=1}{c}})

Synthetic division asks how much of P is left over after taking out x − c; what's left over is just the value P(c).

What the symbols mean

$P(x)$
The polynomial
Here x³ − 4x² + x + 6.
$c$
Divisor's number
Dividing by x − c.
$Q(x)$
Quotient
What's left after dividing P by x − c: one degree lower.
$R$
Remainder
The leftover constant, which always equals P(c).

Synthetic division by $x - c$ works on the coefficients alone: bring down the first, then repeatedly multiply by $c$ and add the next coefficient.
The numbers produced are the coefficients of the quotient $Q(x)$ , and the last one is the remainder $R$ , so $P(x) = (x - c)Q(x) + R$ .
Put $x = c$ into that identity: the first term vanishes, so $R = P(c)$ (the remainder theorem). If $R = 0$ , then $x - c$ is a factor and $c$ is a zero (the factor theorem).
Evaluating this way, $((x - 4)x + 1)x + 6$ , takes one multiplication and one addition per coefficient: Horner's method, the cheapest way to evaluate a polynomial.

Learn the math

Dividing PolynomialsAlgebra · Lesson 34, Polynomial Functions Zeroes of PolynomialsAlgebra · Lesson 35, Polynomial Functions