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
xfrom 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)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) = 0for some value of
ais called a root or a zero of the polynomial.
- Roots are the values of
xfor 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 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-a) is a factor, and
a is a root.