Synthetic Division/Examples
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Examples of Synthetic Division
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:
- $4 x^3 - 2 x^2 + 3 x - 1 = \paren {x - 2} \paren {4 x^2 + 6 x + 15} + 29$