What is the factor theorem for polynomials?

The factor theorem states that if a polynomial f(x) has a factor (x-a), then f(a) = 0.

The factor theorem is a useful tool in algebraic manipulation of polynomials. It allows us to determine whether a given value is a root of a polynomial, and also to factorise polynomials.

To use the factor theorem, we first need to identify a potential factor of the polynomial. For example, if we have the polynomial f(x) = x^3 - 6x^2 + 11x - 6, we might suspect that (x-1) is a factor. To check this, we evaluate f(1) and see if it equals zero:

f(1) = 1^3 - 6(1)^2 + 11(1) - 6 = 0

Since f(1) = 0, we know that (x-1) is a factor of f(x). We can then use polynomial division or synthetic division to find the other factors of f(x). In this case, we get:

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

So the roots of f(x) are x=1, x=2, and x=3.

In summary, the factor theorem is a powerful tool for factorising polynomials and finding their roots. By identifying potential factors and evaluating the polynomial at those values, we can quickly determine whether a given value is a root and factorise the polynomial accordingly.