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

Lucy Kirkham

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Sum & Product of Roots

How do I find the sum & product of roots of polynomials?

  • 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 with n roots alpha subscript 1 comma space alpha subscript 2 comma space... comma space alpha subscript n space

    • The polynomial 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

    • an  is the coefficient of the leading term

    • an-1  is the coefficient of the xn-1 term

      • Be careful: this could be equal to zero

    • a0 is the constant term

      • Be careful: this could be equal to zero

  • 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

    • Comparing coefficients of the xn-1 term and the constant term gives

      • a subscript n minus 1 end subscript equals a subscript n open parentheses negative alpha subscript 1 minus alpha subscript 2 minus... negative alpha subscript n close parentheses

      • a subscript 0 equals 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 blank 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 blank 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

      • both of these formulae are in your formula booklet

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

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

  • Form two equations using the roots

    • One using the sum of the roots formula

    • One using the product of the roots formula

  • Solve for any unknowns

Examiner Tips and Tricks

  • Examiners might trick you by not having an xn-1 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.

a) Use the sum of all the roots to find the value of alpha.

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

b) Use the product of all the roots to 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 STEM

    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.