Polynomial Division (CIE A Level Maths: Pure 3)

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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

2.5.2 Polynomial not polynomial, Edexcel A Level Maths: Pure revision notes 

What is polynomial division?

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

2.5.2 Polynomial Division What Is, Edexcel A Level Maths: Pure revision notes

  • 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:

2.5.2 Bus Stop Div, Edexcel A Level Maths: Pure revision notes

  • 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

2.5.2 Polynomial Division Illustration_1, Edexcel A Level Maths: Pure revision notes

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

2.5.2 Polynomial Division Illustration_2, Edexcel A Level Maths: Pure revision notes

  •  Continue until you are left with zero

2.5.2 Polynomial Division Illustration_3, Edexcel A Level Maths: Pure revision notes

  • 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

2.5.2 Polynomial Division Example, Edexcel A Level Maths: Pure revision notes

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Quadratic Divisor

What are the degrees of the quotient and remainder when a polynomial is divided by a quadratic divisor?

  • Suppose a polynomial of degree n is divided by a quadratic divisor
    • fraction numerator straight f open parentheses x close parentheses over denominator a x squared plus b x plus c end fraction equals straight q open parentheses x close parentheses plus fraction numerator straight r open parentheses x close parentheses over denominator a x squared plus b x plus c end fraction
  • The quotient q will have degree n minus 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

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.