Definition:Synthetic Division

Definition

Synthetic division is a technique for dividing a polynomial by a linear factor.

Let $\map P x$ be a polynomial of degree $n$.

Dividing $\map P x$ by $x - a$ we get:

$\map P x = \paren {x - a} \map Q x + r$

where $\map Q x$ is a polynomial of degree $n - 1$ and $r$ is a constant.

Applying Horner's rule to evaluate $\map P a$, the coefficient $a_n$ together with the intermediate quantities produced as each pair of brackets is removed are the coefficients of $\map q x$.

Hence the final result is $r = \map P a$.

Examples

Arbitrary Example

Let $\map P x$ be the cubic:

$\map P x = 4 x^3 - 2 x^2 + 3 x - 1$

Applying Horner's rule:

$\map P x = \paren {\paren {4 x - 2} x + 3} x - 1$

Let $x = 2$.

Then:

\(\ds 4 x - 2\)

\(=\)

\(\ds 4 \times 2 - 2\)

\(\ds = 6\)

\(\ds 6 x + 3\)

\(=\)

\(\ds 6 \times 2 + 3\)

\(\ds = 15\)

\(\ds 15 x - 1\)

\(=\)

\(\ds 15 \times 2 - 1\)

\(\ds = 29\)

Noting that the coefficient of $x^3$ is $3$, we obtain:

$4 x^3 - 2 x^2 + 3 x - 1 = \paren {x - 2} \paren {4 x^2 + 6 x + 15} + 29$

Also see

• Definition:Horner's Rule

• Results about synthetic division can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Horner's method: 1.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): synthetic division
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Horner's method: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): synthetic division