Cambridge O Level Additional Mathematics · Syllabus 4037 · Factors of Polynomials
Polynomial Long Division
What is Polynomial Long Division?
A column method for dividing one polynomial by another, laid out like numerical long division with the columns representing descending powers of x. At each stage the leading term of what remains is divided by the leading term of the divisor to give the next term of the quotient, that term is multiplied through the divisor, and the product is subtracted. Any power missing from the dividend must be written in with a zero coefficient so that the columns stay aligned. When the divisor is a factor, the process ends with a remainder of zero and the quotient is the cofactor.
This definition is part of the Factors of Polynomials chapter in Cambridge O Level Additional Mathematics.
Polynomial Long Division 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 Polynomial Long Division
Why do you write \(0x^2\) in polynomial long division?
Because long division is a column method and each column is a power of \(x\). If the dividend has a missing power, such as \(x^3-7x+6\) with no \(x^2\) term, you must insert \(0x^2\) and write \(x^3+0x^2-7x+6\), otherwise the terms slip into the wrong columns and every later line is wrong. At each step divide the leading terms, multiply through the divisor, then subtract, remembering that subtracting a negative term adds.