Factor Theorem
What is the factor theorem?
- The factor theorem is used to find the linear factors of polynomial equations
- This topic is closely tied to finding the zeros and roots of a polynomial function/equation
- As a rule of thumb a zero refers to the polynomial function and a root refers to a polynomial equation
- For any polynomial function P(x)
- (x - k) is a factor of P(x) if P(k) = 0
- P(k) = 0 if (x - k) is a factor of P(x)
How do I use the factor theorem?
- Consider the polynomial function P(x) = anxn + an-1xn-1 + … + a1x + a0 and (x - k) is a factor
- Then, due to the factor theorem P(k) = ankn + an-1kn-1 + … + a1k + a0 = 0
- , where Q(x) is a polynomial that is a factor of P(x)
- Hence, , where Q(x) is another factor of P(x)
- If the linear factor has a coefficient of x then you must first factorise out the coefficient
- If the linear factor is
Examiner Tip
- A common mistake in exams is using the incorrect sign for either the root or the factor
- If you are asked to find integer solutions to a polynomial then you only need to consider factors of the constant term
Worked example
Determine whether is a factor of the following polynomials:
It is given that is a factor of .