Cambridge O Level Additional Mathematics · Syllabus 4037 · Factors of Polynomials
Remainder Theorem
What is Remainder Theorem?
A result stating that when a polynomial P(x) is divided by a linear divisor x - a, the remainder is the single number P(a), obtained by substituting x = a into the polynomial. It converts a division problem into an evaluation problem, so the remainder can be found without carrying out the division. For a divisor written in another form, such as x + 2 or 2x - 1, the value substituted is the value that makes the divisor zero, namely -2 and one half respectively.
This definition is part of the Factors of Polynomials chapter in Cambridge O Level Additional Mathematics.
Remainder Theorem in context
The remainder theorem states that when a polynomial \(P(x)\) is divided by the linear divisor \(x-a\), the remainder is the single number \(P(a)\). It turns a slow division into a fast substitution: to find the remainder you set the divisor equal to zero, take that test value, and evaluate the polynomial at it. For a divisor such as \(x+2\) the test value is \(x=-2\), and for \(2x-1\) it is \(x=\tfrac12\). The whole of Chapter 3 is built on this one connection between dividing and substituting.
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 Remainder Theorem
What does the remainder theorem say?
The remainder theorem says that the remainder when a polynomial \(P(x)\) is divided by \(x-a\) is \(P(a)\). You do not need to divide: set the divisor equal to zero to get the test value, then substitute it. For \(P(x)=2x^3-5x^2+4x-7\) divided by \(x-2\), the remainder is \(P(2)=-3\). For a divisor such as \(3x+2\), the test value is \(x=-\tfrac23\).
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.