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Sum & Product of Roots of Polynomials (DP IB Analysis & Approaches (AA)): Revision Note

Lucy Kirkham

Written by: Lucy Kirkham

Reviewed by: Dan Finlay

Updated on

Sum & product of roots

How do I find the sum & product of the roots of a polynomial?

  • Suppose  P open parentheses x close parentheses equals a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus blank horizontal ellipsis plus a subscript 1 x plus a subscript 0 is a polynomial of degree n

    • a subscript n  is the coefficient of the leading term

    • a subscript n minus 1 end subscript  is the coefficient of the x to the power of n minus 1 end exponent term

      • This could be equal to zero

      • e.g. P open parentheses x close parentheses equals 3 x to the power of 4 plus x squared minus x plus 1

    • a subscript 0 is the constant term

      • This could be equal to zero

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

  • The n roots of the equation P open parentheses x close parentheses equals 0 are denoted as alpha subscript 1 comma space alpha subscript 2 comma space... comma space alpha subscript n space

    • Some roots might be complex and/or repeated

    • You can find their sum and product without finding the values of the roots

Examiner Tips and Tricks

The equation P open parentheses x close parentheses equals 0 is written as sum from r equals 0 to n of a subscript r x to the power of r equals 0 comma space a subscript n not equal to 0 in the formula booklet.

  • In factorised form P left parenthesis x right parenthesis equals a subscript n open parentheses x minus alpha subscript 1 close parentheses open parentheses x minus alpha subscript 2 close parentheses... open parentheses x minus alpha subscript n close parentheses

    • The coefficient of the x to the power of n minus 1 end exponent term is a subscript n open parentheses negative alpha subscript 1 minus alpha subscript 2 minus... negative alpha subscript n close parentheses

    • The constant term is a subscript n open parentheses negative alpha subscript 1 close parentheses cross times open parentheses negative alpha subscript 2 close parentheses cross times... cross times open parentheses negative alpha subscript n close parentheses

  • The sum of the roots is given by:

    •  alpha subscript 1 plus alpha subscript 2 plus horizontal ellipsis plus alpha subscript n equals negative a subscript n minus 1 end subscript over a subscript n

  • The product of the roots is given by:

    • alpha subscript 1 cross times blank alpha subscript 2 cross times horizontal ellipsis cross times alpha subscript n equals fraction numerator open parentheses negative 1 close parentheses to the power of n a subscript 0 over denominator a subscript n end fraction

Examiner Tips and Tricks

Both of these formulas are in your formula booklet.

For example, consider 5 x to the power of 4 plus 2 x cubed minus 3 x squared plus x minus 7 equals 0

  • The sum of the roots is equal to negative 2 over 5

  • The product of the roots is equal to fraction numerator open parentheses negative 1 close parentheses to the power of 4 cross times open parentheses negative 7 close parentheses over denominator 5 end fraction equals negative 7 over 5

How can I find unknowns if I am given the sum and/or product of the roots of a polynomial?

  • Write down all the roots you know

    • If you know a complex root of a real polynomial then its complex conjugate is another root

  • You can form two equations using the roots

    • One using the sum of the roots formula

    • One using the product of the roots formula

  • Solve the equations to find any unknowns

Examiner Tips and Tricks

Examiners might trick you by not having an x to the power of n minus 1 end exponent term or a constant term.

To make sure you do not get tricked, you can write out the full polynomial using 0 as a coefficient where needed. For example, write x to the power of 4 plus 2 x squared minus 5 x as x to the power of 4 plus 0 x cubed plus 2 x squared minus 5 x plus 0.

Worked Example

2 minus 3 straight i, 5 over 3 straight i and alpha are three roots of the equation 18 x to the power of 5 minus 9 x to the power of 4 plus 32 x cubed plus 794 x squared minus 50 x plus k equals 0, where k is a real constant.

a) Given that alpha is a real number, find the value of alpha.

2-7-4-ib-aa-hl-sum-product-roots-a-we-solution

b) Find the value of k.

2-7-4-ib-aa-hl-sum-product-roots-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.