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`.

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)`

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)

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