Roots and Factor Theorem

What is a Root?

A root (or zero) of a polynomial is a value that makes the polynomial equal to zero.

For $p(x) = x^2 - 5x + 6$, the roots are the solutions to:

$x^2 - 5x + 6 = 0$

We can factor this: $(x - 2)(x - 3) = 0$

So the roots are $x = 2$ and $x = 3$.

Check: Plug them back in:

The Factor Theorem

This is the key insight connecting roots and factors:

If $r$ is a root of $p(x)$, then $(x - r)$ is a factor of $p(x)$.

And vice versa:

If $(x - r)$ is a factor of $p(x)$, then $r$ is a root.

Example:

• If $x = 4$ is a root of $p(x)$, then $(x - 4)$ is a factor.
• If $(x + 2)$ is a factor of $p(x)$, then $x = -2$ is a root.

Why is This Useful?

Finding factors from roots:

If you can find a root, you’ve found a factor.

Finding roots from factors:

Once factored, roots are obvious.

Reducing polynomials:

Once you find one root $r$, divide out $(x - r)$ and work with a smaller polynomial.

Example: Factoring a Cubic

Factor: $x^3 - 6x^2 + 11x - 6$

Step 1: Find a root

Try small integers: $\pm 1, \pm 2, \pm 3, \ldots$

Try $x = 1$:

$1 - 6 + 11 - 6 = 0$ ✓

So $x = 1$ is a root, which means $(x - 1)$ is a factor.

Step 2: Divide out the factor

Divide $x^3 - 6x^2 + 11x - 6$ by $(x - 1)$.

Result: $x^2 - 5x + 6$

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

Step 3: Factor the remaining quadratic

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

Final answer:

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

Roots: $x = 1, 2, 3$