Factor Theorem

## Factor Theorem

The Factor Theorem simply states that if you divide a polynomial f(x) by a binomial (x-c), and if (x-c) is a factor, then the division will have no remainder.

It also means that f(c) = 0.

An example shows how this works. f(x) = x^2 + x - 6 factorises to (x - 2) and (x - 3). Dividing f(x) by (x - 2) gives the answer of (x - 3), which is another factor but with no remainder.

And when x=2, the function evaluates to zero: f(2) = (2 - 2)(2 - 3) = 0.

## Example 1

Determine the factors of f(x) = x^2 - 3x - 28 given that f(-4) = f(7) = 0

Use the factor theorem

If f(-4) = 0 then (x + 4) is a factor

Similarly if f(7) = 0, then (x-7) is a factor.

Answer: (x + 4) and (x - 7)

## Example 2

Is (x - 3) a factor of 12x^3 - 32x^2 - 85x + 175?

If (x - 3) is a factor, then f(3) should evaluate to zero (factor theorem).

Answer: No. f(3) = 12(3)^3 - 32(3)^2 - 85(3) + 175 = -44 (Factor theorem)