IBMaths: AA HLDPRevision Notes2. Functions2.7 Polynomial Functions2.7.4 Roots of Polynomials

Roots of Polynomials (DP IB Maths: AA HL)

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Lucy

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Head of STEM

Sum & Product of Roots

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

Suppose $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ is a polynomial of degree n with n roots $α_{1}, α_{2}, . . ., α_{n}$
- The polynomial is written as $\sum_{r = 0}^{n} a_{r} x^{r} = 0, a_{n} \neq 0$ in the formula booklet
- a_n is the coefficient of the leading term
- a_n_-1 is the coefficient of the xⁿ^-1 term
  - Be careful: this could be equal to zero
- a₀is the constant term
  - Be careful: this could be equal to zero
In factorised form: $P (x) = a_{n} (x - α_{1}) (x - α_{2}) . . . (x - α_{n})$
- Comparing coefficients of the xⁿ^-1 term and the constant term gives
  - $a_{n - 1} = a_{n} (- α_{1} - α_{2} - . . . - α_{n})$
  - $a_{0} = a_{n} (- α_{1}) \times (- α_{2}) \times . . . \times (- α_{n})$
The sum of the roots is given by:
- $α_{1} + α_{2} + \dots + α_{n} = - \frac{a_{n - 1}}{a_{n}}$
The product of the roots is given by:
- $α_{1} \times α_{2} \times \dots \times α_{n} = \frac{{(- 1)}^{n} a_{0}}{a_{n}}$
  - 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

Exam Tip

Examiners might trick you by not having an x^n-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 $x^{4} + 2 x^{2} - 5 x$ as $x^{4} + 0 x^{3} + 2 x^{2} - 5 x + 0$

Worked example

$2 - 3 i$ , $\frac{5}{3} i$ and $α$ are three roots of the equation $18 x^{5} - 9 x^{4} + 32 x^{3} + 794 x^{2} - 50 x + k = 0$ .

Use the sum of all the roots to find the value of

α

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

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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Author: Lucy

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. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.