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Polynomial Division (DP IB Analysis & Approaches (AA)): Revision Note

Lucy Kirkham

Written by: Lucy Kirkham

Reviewed by: Dan Finlay

Updated on

Polynomial division

What is polynomial division?

  • Polynomial division is the process of dividing two polynomials

    • fraction numerator P open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction

      • D open parentheses x close parentheses is called the divisor

      • The degree of the divisor is less than or equal to the degree of the polynomial

  • The result gives a quotient polynomial Q open parentheses x close parentheses and a remainder polynomial R open parentheses x close parentheses

    • fraction numerator P open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction equals Q open parentheses x close parentheses plus fraction numerator R open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction

What are the degrees of the quotient and remainder?

  • If P left parenthesis x right parenthesis has degree n and is divided by a divisor D left parenthesis x right parenthesis with degree k space less or equal than space n

    •  fraction numerator P open parentheses x close parentheses over denominator D left parenthesis x right parenthesis end fraction equals Q open parentheses x close parentheses plus fraction numerator R left parenthesis x right parenthesis over denominator D open parentheses x close parentheses end fraction

  • The degree of the quotient Q left parenthesis x right parenthesis is equal to n minus k

  • The degree of the remainder R left parenthesis x right parenthesis is less than k

  • For example, when x to the power of 5 minus 7 is divided by x squared plus 2

    • The degree of the quotient is 3

    • The degree of the remainder is less than 2

      • It could be 0 (a constant term) or 1 (a linear expression)

How do I divide polynomials?

  • Let's use the example:

    • P open parentheses x close parentheses equals 2 x to the power of 4 plus 3 x cubed minus x squared plus 5

    • D open parentheses x close parentheses equals x squared plus 2 x minus 1

  • STEP 1
    Divide the leading term of the polynomial P left parenthesis x right parenthesis by the leading term of the divisor

    • This is the first term of the quotient

    • e.g. fraction numerator 2 x to the power of 4 over denominator x squared end fraction equals 2 x squared

  • STEP 2
    Multiply the divisor by this term

    • e.g. 2 x squared open parentheses x squared plus 2 x minus 1 close parentheses equals 2 x to the power of 4 plus 4 x cubed minus 2 x squared

    Subtract this from the original polynomial P left parenthesis x right parenthesis to find the current remainder

    • The leading term should be cancelled out

      • e.g. open parentheses 2 x to the power of 4 plus 3 x cubed minus x squared plus 5 close parentheses minus open parentheses 2 x to the power of 4 plus 4 x cubed minus 2 x squared close parentheses equals negative x cubed plus x squared plus 5

  • STEP 3
    Repeat steps 1 – 2 using the current remainder as the main polynomial

    • Keep repeating the steps until the degree of the remainder is less than the degree of the division

    • Find the second term of the quotient

      • e.g. negative x cubed over x squared equals negative x

      • e.g. negative x open parentheses x squared plus 2 x minus 1 close parentheses equals negative x cubed minus 2 x squared plus x

      • e.g. open parentheses negative x cubed plus x squared plus 5 close parentheses minus open parentheses negative x cubed minus 2 x squared plus x close parentheses equals 3 x squared minus x plus 5

    • Find the third term of the quotient

      • e.g. fraction numerator 3 x squared over denominator x squared end fraction equals 3

      • e.g. 3 open parentheses x squared plus 2 x minus 1 close parentheses equals 3 x squared plus 6 x minus 3

      • e.g. open parentheses 3 x squared minus x plus 5 close parentheses minus open parentheses 3 x squared plus 6 x minus 3 close parentheses equals negative 7 x plus 8

  • STEP 5
    Identify the quotient and the remainder

    • The quotient is the sum of all the terms from step 1

      • e.g. Q open parentheses x close parentheses equals 2 x squared minus x plus 3

    • The remainder is the last remainder from step 2

      • e.g. R open parentheses x close parentheses equals negative 7 x plus 8

Examiner Tips and Tricks

There are multiple ways to set out polynomial division, such as using a bus stop or a grid. The steps above are used in both methods. You can see an example of the bus stop method in the worked example.

How do I divide by comparing coefficients?

  • STEP 1
    Write the expression as P open parentheses x close parentheses equals Q open parentheses x close parentheses D open parentheses x close parentheses plus R open parentheses x close parentheses

    • Use the facts about the degrees to get the correct number of terms

      • e.g. 2 x to the power of 4 plus 3 x cubed minus x squared plus 5 equals open parentheses x squared plus 2 x minus 1 close parentheses open parentheses a x squared plus b x plus c close parentheses plus open parentheses d x plus f close parentheses

  • STEP 2
    Work out the leading coefficient of the polynomial on the right-hand side and set it equal to the leading coefficient on the left-hand side

    • You can find the leading term of the quotient

      • e.g. for x to the power of 4: 2 equals a therefore a equals 2

  • STEP 3
    Repeat the step for the next leading term

    • You might have to use the previous value

      • e.g. for x cubed: 3 equals b plus 2 a squared equals b plus 4 therefore b equals negative 1

  • STEP 4
    Keep repeating to find all the unknowns

    • Remember to include missing terms such as 0 x

      • e.g. for x squared: negative 1 equals c plus 2 b minus a equals c minus 2 minus 2 equals c minus 4 therefore c equals 1

      • e.g. for x: 0 equals 2 c minus b plus d equals 6 plus 1 plus d equals 7 plus d therefore d equals negative 7

      • e.g. for constant terms: 5 equals negative c plus f equals negative 3 plus f therefore f equals 8

Examiner Tips and Tricks

In an exam you can use whichever method to divide polynomials - just make sure your method is written clearly so that if you make a mistake you can still get a mark for your method!

Worked Example

a) Perform the division fraction numerator x to the power of 4 plus 11 x squared minus 1 over denominator x plus 3 end fraction. Hence write x to the power of 4 plus 11 x squared minus 1 in the form Q left parenthesis x right parenthesis cross times left parenthesis x plus 3 right parenthesis plus R.

2-7-2-ib-aa-hl-polynomial-division-a-we-solution-1-2
2-7-2-ib-aa-hl-polynomial-division-a-we-solution-2-2

b) Find the quotient and remainder for fraction numerator x to the power of 4 plus 4 x cubed minus x plus 1 over denominator x squared minus 2 x end fraction. Hence write x to the power of 4 plus 4 x cubed minus x plus 1 in the form Q left parenthesis x right parenthesis cross times left parenthesis x squared minus 2 x right parenthesis plus R left parenthesis x right parenthesis.

2-7-2-ib-aa-hl-polynomial-division-b-we-solution-

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    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of Content Creation

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.