Polynomial Division (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Polynomial division

What is polynomial division?

• Polynomial division is the process of dividing two polynomials

  • This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator

• Polynomial division is a method for splitting polynomials into factor pairs

  • (with or without a remainder term)

• The main uses of polynomial division are

  • factorising polynomials

  • simplifying 'top-heavy' algebraic fractions

How do I divide polynomials?

• The method used for polynomial division is just like the long division method for numbers

  • sometimes called the 'bus stop method'

• The answer to a polynomial division question is built up term by term

  • Start by dividing by the highest power term

  • Write out this multiplied by the divisor and subtract

•  Continue to divide by each reducing power term and subtracting your answer each time

•  Continue until you are left with zero

• If the divisor is not a factor of the polynomial then there will be a remainder term left at the end of the division

Quadratic divisor

What is meant by a quadratic divisor?

• Polynomial division usually involves dividing by a linear term

  • a term of the form $\left(x+p\right)$ where $p$ is a constant and usually an integer

• It is possible to divide a polynomial by a quadratic term (and cubic, etc)

  • this would be a term of the form $\left(x^2+qx+r\right)$ where $q$ and $r$ are constants

  • this is what is meant by a quadratic divisor

How do I divide by a quadratic divisor?

• The process is the same as for a linear divisor

  • However, as $x^2$ will not divide into $x$ (in the polynomial division sense at least) the remainder, if there is one, could be of linear form, i.e. $\left(rx+s\right)$ where $r$ and $s$ are constants

    • It is possible that $r=0$ and so the remainder is still a constant