Use of the Factor Theorem for Rational Values of the Variable for Polynomials

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.