Polynomial Division

Why Divide Polynomials?

From the factor theorem: if $x = 1$ is a root of $x^3 - 6x^2 + 11x - 6$, then $(x - 1)$ is a factor.

But what’s the other factor? We divide to find out:

$(x^3 - 6x^2 + 11x - 6) \div (x - 1) = \text{ ?}$

Long Division

Works just like long division with numbers.

The process:

1. Divide the leading terms: $x^3 \div x = x^2$
2. Multiply: $x^2 \times (x - 1) = x^3 - x^2$
3. Subtract and bring down
4. Repeat until done

Result:

$x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6)$

Synthetic Division

A shortcut when dividing by $(x - r)$.

For $(x - 1)$, we use $r = 1$.

Setup: Write the coefficients of the dividend.

$x^3 - 6x^2 + 11x - 6 \quad \rightarrow \quad 1, -6, 11, -6$

Process:

How it works:

1. Bring down the first coefficient: $1$
2. Multiply by $r$, add to next: $-6 + 1 = -5$
3. Multiply by $r$, add to next: $11 + (-5) = 6$
4. Multiply by $r$, add to next: $-6 + 6 = 0$ (remainder)

Reading the result:

Coefficients $1, -5, 6$ with remainder $0$

$\rightarrow \quad x^2 - 5x + 6 \quad \text{remainder } 0$

When to Use Each Method

Synthetic division is faster, but only works for linear divisors.