Improper Algebraic Fractions (Edexcel IAL Maths)

Spec point — 1.1

Improper algebraic fractions

What are improper algebraic fractions?

• The degree of the numerator is greater than or equal to the degree of the denominator

How do I simplify improper algebraic fractions?

• Write as a quotient and a remainder
• The algebraic equivalent of changing a top-heavy fraction to a mixed number

Exam Hint Remember that simple cases are sometimes the hardest to spot!

Worked Example

Quadratic Divisor

Spec point — 1.1

What is a quadratic divisor?

• A polynomial f(x) of degree $n$ divided by a quadratic divisor takes the form

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

• 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 still 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