Roots of Polynomials | DP IB Maths: AA HL Revision Notes 2021

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

Examiner 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

Roots of Polynomials (DP IB Maths: AA HL)

Revision Note

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

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

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs