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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 1. Number & Algebra / 1.9 Further Complex Numbers / 1.9.3 Complex Roots of Polynomials


1.9.3 Complex Roots of Polynomials


Complex Roots of Quadratics

What are complex roots?

  • A quadratic equation can either have two real roots (zeros), a repeated real root or no real roots
    • This depends on the location of the graph of the quadratic with respect to the x-axis
  • If a quadratic equation has no real roots we would previously have stated that it has no real solutions
    • The quadratic equation will have a negative discriminant
    • This means taking the square root of a negative number
  • Complex numbers provide solutions for quadratic equations that have no real roots

How do we solve a quadratic equation when it has complex roots?

  • If a quadratic equation takes the form ax2 + bx + c = 0 it can be solved by either using the quadratic formula or completing the square
  • If a quadratic equation takes the form ax2 + b = 0 it can be solved by rearranging
  • The property i = √-1 is used
    • square root of negative a end root equals square root of a cross times negative 1 end root equals square root of a cross times square root of negative 1 end root
  • If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs
    • If z = m + ni (n ≠ 0) is a root of a quadratic with real coefficients then z* = m - ni is also a root
  • The real part of the solutions will have the same value as the x coordinate of the turning point on the graph of the quadratic
  • When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
    • To solve these you can use the quadratic formula

How do we factorise a quadratic equation if it has complex roots?

  • If we are given a quadratic equation in the form az2 + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0 we can use its complex roots to write it in factorised form
    • Use the quadratic formula to find the two roots, z  = p + qi and z* = p - qi
    • This means that z – (p + qi) and z – (p – qi) must both be factors of the quadratic equation
    • Therefore we can write az2 + bz + c = (z – (p + qi))( z – (p - qi))
    • This can be rearranged into the form (z – p – qi)(z – p + qi)

How do we find a quadratic equation when given a complex root?

  • If we are given a complex root in the form z = p + qi we can find the quadratic equation in the form az2 + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0
    • We know that the second root must be z* = p - qi
    • This means that z – (p + qi) and z – (p – qi) must both be factors of the quadratic equation
    • Therefore we can write az2 + bz + c = (z – (p + qi))( z – (p - qi))
    • Rewriting this as ((z – p) – qi))((z – p) + qi)) makes expanding easier
    • Expanding this gives the quadratic equation z2 – 2pz + (p2 + q2)
      • a = 1
      • b = -2p
      • c = p2 + q2
  • This demonstrates the important property (x – z)(x – z*) = x2 - 2Re(z)x + |z|2

Exam Tip

  • Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation
    • You should get 0 if correct! [Note: 0 is equivalent to 0 plus 0 bold i]

Worked Example

a)
Solve the quadratic equation z2 - 2z + 5 = 0 and hence, factorise z2 - 2z + 5.

1-9-3-ib-aa-hl-complex-roots-we-solution-1-a

b)
Given that one root of a quadratic equation is z = 2 – 3i, find the quadratic equation in the form az2 + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0.

1-9-3-ib-aa-hl-complex-roots-we-solution-1-b

Complex Roots of Polynomials

How many roots should a polynomial have?

  • We know that every quadratic equation has two roots (not necessarily distinct or real)
  • This is a particular case of a more general rule:
    • Every polynomial equation, with real coefficients, of degree n has n roots
    • The n roots are not necessarily all distinct and therefore we need to count any repeated roots that may occur individually
  • From the above rule we can state the following:
    • A cubic equation of the form a x cubed plus b x squared plus c x plus d equals 0 can have either:
      • 3 real roots
      • Or 1 real root and a complex conjugate pair
  • A quartic equation of the form a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 will have one of the following cases for roots:
    • 4 real roots
    • 2 real and 2 nonreal (a complex conjugate pair)
    • 4 nonreal (two complex conjugate pairs)
  • When a real polynomial of any degree has one complex root it will always also have the complex conjugate as a root

How do we solve a cubic equation with complex roots?

  • Steps to solve a cubic equation with complex roots
    • If we are told that p + qi is a root, then we know p - qi is also a root
    • This means that z – (p + qi) and z – (p – qi) must both be factors of the cubic equation
    • Multiplying the above factors together gives us a quadratic factor of the form (Az2 + Bz + C)
    • We need to find the third factor (z - α)
    • Multiply the factors and equate to our original equation to get
      • open parentheses A z squared plus B z plus C close parentheses open parentheses z minus alpha close parentheses equals a x cubed plus b x squared plus c x plus d
    • From there either
      • Expand and compare coefficients to find
      • Or use polynomial division to find the factor (z - α)
    • Finally, write your three roots clearly

How do we solve a polynomial of any degree with complex roots?

  • When asked to find the roots of any polynomial when we are given one, we use almost the same method as for a cubic equation
    • State the initial root and its conjugate and write their factors as a quadratic factor (as above) we will have two unknown roots to find, write these as factors (z - α) and (z - β)
    • The unknown factors also form a quadratic factor (z - α)(z - β)
    • Then continue with the steps from above, either comparing coefficients or using polynomial division
      • If using polynomial division, then solve the quadratic factor you get to find the roots α and β

How do we solve polynomial equations with unknown coefficients?

  • Steps to find unknown variables in a given equation when given a root:
    • Substitute the given root p + qi into the equation f(z) = 0
    • Expand and group together the real and imaginary parts (these expressions will contain our unknown values)
    • Solve as simultaneous equations to find the unknowns
    • Substitute the values into the original equation
    • From here continue using the previously described methods for finding other roots for the polynomial

How do we factorise a polynomial when given a complex root?

  • If we are given a root of a polynomial of any degree in the form z  = p + qi
    • We know that the complex conjugate, z* = p – qi is another root
    • We can write (z – (p + qi)) and ( z – (p - qi)) as two linear factors
      • Or rearrange into one quadratic factor
    • This can be multiplied out with another factor to find further factors of the polynomial
  • For higher order polynomials more than one root may be given
    • If the further given root is complex then its complex conjugate will also be a root
    • This will allow you to find further factors  

Exam Tip

  • You can speed up multiplying two complex conjugate factors together by
    • rewrite (z – (p + qi))(z – (p - qi)) as ((z – p) - qi))( (z – p) + qi))
    • Then ((z – p) - qi))( (z – p) + qi)) = (z - p)2 - (qi)2 = (z - p)2 + q2
  • If you are working on a calculator paper read the question carefully to see how much of the working needs to be shown but always remember to use your GDC to check your working where you can

Worked Example

Given that one root of a polynomial p(x) = z3 + z2 – 7z + 65 is 2 – 3i, find the other roots.

1-9-3-ib-aa-hl-complex-roots-we-solution-2



  • 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
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
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                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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