Polynomial Division (Cambridge International A-Level Mathematics)

Polynomial division

What is a polynomial?

• A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only

What is polynomial division?

• Polynomial division is a method for splitting polynomials into factor pairs (with or without an accompanying remainder term)

• At A level you will most frequently use it to factorise polynomials, or when dealing with improper (ie 'top-heavy') algebraic fractions

How do I divide polynomials?

• The method used for polynomial division is just like the long division method (sometimes called 'bus stop division') used to divide regular numbers:

• At A level you will normally be dividing a polynomial dividend of degree 3 or 4 by a divisor in the form (x ± p)

• The answer to a polynomial division question is built up term by term, working downwards in powers of the variable (usually x)

• 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

Worked Example

Quadratic Divisor

What is a quadratic divisor?

• Suppose a polynomial of degree $n$ is divided by a quadratic divisor

  • $\frac{ \mathrm{f} ( x )}{a x^{2} + b x + c} = \mathrm{q} ( x ) + \frac{ \mathrm{r} ( x )}{a x^{2} + b x + c}$

• The quotient q will have degree $n-2$

• The degree of the remainder r will be less than $2$

  • It could be degree 1 (linear)
  • Or it could be degree 0 (constant)

How do I divide a polynomial by a quadratic divisor?

• You use polynomial division!

• Step 1 Divide the leading term of the polynomial by the squared term of the divisor

  • This gives the leading term of the quotient

• Step 2 Multiply this term by the divisor

• Step 3 Subtract this from the polynomial to get a new polynomial with a lower degree

• Continue these steps until you have an expression with a degree lower than 2