Polynomial Division (Cambridge International IGCSE Further Maths)

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

Worked Example For the polynomial $\mathrm{f} (x)=x^4+11x²−1$divide $\mathrm{f} (x)$ by $x+3$ and write the remainder.

Set up the polynomial division ('bus stop')

There is no $x^{3}$ term so write this as $0x^{3}$ in the method. There is no $x$ term so write this as $0x$ in the method.

The first division step to consider is $x^{4}\div x$.

.Multiply $x^{3}$ by $x+3$ and subtract from $x^{4}+0x$.

Bring the $11x^{2}$ down and divide $-3x^{3}$ by $x$. Continue with each step until you are finished.

Final answer: The remainder is 179.

Quadratic Divisor

What is meant by a quadratic divisor?

• Polynomial division usually involves dividing by a linear term

  • a term of the form (x+p) 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 (x²+q x+r) 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. (r x+s) where $r$ and $s$ are constants

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

Exam Hint

• Give yourself plenty of room to do polynomial division

  • Not only will this help avoid errors, it will make your working clear

• If you make a mistake and change something, fine, but if your method starts to get too messy it is best to restart

Worked Example Find the remainder when $x^{4}+4x^{3}-x+1$ is divided by $x^{2}-2x$.

Set up the polynomial division ('bus stop') - there is no $x^{2}$ term so write this as $0x^{2}$ in the method.

The first division step to consider is $x^{4}\div x^{2}$.

Final answer: The remainder is$23x+1$.