Cambridge O Level Additional Mathematics · Syllabus 4037 · Factors of Polynomials
Factor Theorem
What is Factor Theorem?
A result stating that x - a is a factor of the polynomial P(x) if and only if P(a) = 0. It is the special case of the remainder theorem in which the remainder is zero, so the division leaves nothing over and P(x) can be written as x - a multiplied by another polynomial. The equivalence runs both ways: a zero value proves a factor, and a factor forces the value to be zero, so a non-zero value disproves the factor outright.
This definition is part of the Factors of Polynomials chapter in Cambridge O Level Additional Mathematics.
Factor Theorem in context
The factor theorem is the special case of the remainder theorem in which the remainder is zero: \(x-a\) is a factor of \(P(x)\) if and only if \(P(a)=0\). It works in both directions, so a zero value proves a factor and a non-zero value rules one out. Once a linear factor of a cubic is found, dividing it out by polynomial long division or by comparing coefficients leaves a quadratic, which is then factorised or solved by formula to give every root of the equation \(P(x)=0\).
Questions students ask about Factor Theorem
When do you use the remainder theorem and when the factor theorem?
Use the remainder theorem when a question asks for the remainder, or gives a remainder and asks for an unknown coefficient: the remainder on dividing by \(x-a\) is \(P(a)\). Use the factor theorem when a question mentions a factor or a root: \(x-a\) is a factor if and only if \(P(a)=0\). The factor theorem is simply the remainder theorem with the remainder equal to zero, so a remainder of \(5\) tells you the divisor is not a factor.