What is the factor theorem?
- The factor theorem is a very useful result about polynomials
- A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only
- At A level you will most frequently use the factor theorem as a way to simplify the process of factorising polynomials
What do I need to know about the factor theorem?
- For a polynomial f(x) the factor theorem states that:
- If f(p) = 0, then (x - p) is a factor of f(x)
- If (x - p) is a factor of f(x), then f(p) = 0
- In an exam, the values of p you need to find that make f(p) = 0 are going to be integers close to zero.
- Try p = 1 and -1 first, then 2 and -2, then 3 and -3.
- It is very unlikely that you'll have to go beyond that.
What is the remainder theorem?
- The factor theorem is actually a special case of the more general remainder theorem
- The remainder theorem states that when the polynomial f(x) is divided by (x - a) the remainder is f(a)
- You may see this written formally as f(x) = (x - a)Q(x) + f(a)
- In polynomial division
- Q(x) would be the result (at the top) of the division (the quotient)
- f(a) would be the remainder (at the bottom)
- (x - a) is called the divisor
- In the case when f(a) = 0, f(x) = (x - a)Q(x) and hence (x - a) is a factor of f(x)– the factor theorem!
How do I solve problems involving the remainder theorem?
- If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full
- e.g. The remainder from is
- This is because if f(x) = x2- 2x and a = 3
- If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials
- g. The remainder from is 8 so the value of p can be found by solving , leading to
- In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving
- The more general version of remainder theorem is if f(x) is divided by (ax - b) then the remainder is
- The shortcut is still to evaluate the polynomial at the value of x that makes the divisor (ax - b) zero but it is not necessarily an integer
- Exam questions will use formal mathematical language which can make factor and remainder theorem questions sound more complicated than they are.
- Ensure you are familiar with the various terms from these revision notes