# The Factor Theorem

## Understanding the Factor Theorem

• The Factor Theorem is a powerful tool in algebra which allows us to determine if a given expression is a factor of a polynomial.
• Essentially, the theorem states that a polynomial `f(x)` has a factor `(x-a)` if and only if `f(a) = 0`.

## Applying the Factor Theorem

• To apply the Factor Theorem, plug in the value of `x` from the possible factor `(x-a)` into the polynomial.
• If the answer is zero, then `(x-a)` is a factor of the polynomial.

## Factor Theorem and Division

• The connection to division is that if `(x-a)` is a factor of the polynomial `f(x)`, when `f(x)` is divided by `(x-a)`, the remainder will be zero.
• One can therefore think of the Factor Theorem as a quick way of performing polynomial division.

## Factor Theorem and Roots

• Note that if `f(a) = 0` for some value of `a`, then `a` is called a root or a zero of the polynomial.
• Roots are the values of `x` for which the value of the polynomial is zero.
• There exists a direct relationship between the roots of a polynomial, and the factors of a polynomial.

## Synthetic Division

• Synthetic division is a shorthand method of performing long polynomial division. It’s notably used with the Factor theorem.
• In synthetic division, coefficients are used instead of variables, which simplifies the division process significantly.

Most importantly, the Factor Theorem provides a simple and effective strategy to factorise polynomials, especially those of higher degree. Just as importantly, it gives clues about the graph of the polynomial: wherever the polynomial touches or crosses the `x` axis, `(x-a)` is a factor, and `a` is a root.