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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 2. Functions / 2.7 Polynomial Functions / 2.7.2 Polynomial Division


2.7.2 Polynomial Division


Polynomial Division

What is polynomial division?

  • Polynomial division is the process of dividing two polynomials
    • This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator
  • To do this we use an algorithm similar to that used for division of integers
  • To divide the polynomial P left parenthesis x right parenthesis 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... plus a subscript 1 x plus a subscript 0 by the polynomial D left parenthesis x right parenthesis equals b subscript k x to the power of k plus b subscript k minus 1 end subscript x to the power of k minus 1 end exponent plus... plus b subscript 1 x plus b subscript 0  where k ≤ n
    • STEP 1
      Divide the leading term of the polynomial P(x) by the leading term of the divisor D(x) : fraction numerator a subscript n x to the power of n over denominator b subscript b x to the power of k end fraction equals q subscript m x to the power of m
    • STEP 2
      Multiply the divisor by this term: D left parenthesis x right parenthesis cross times q subscript m x to the power of m
    • STEP 3
      Subtract this from the original polynomial P(x) to cancel out the leading term: R left parenthesis x right parenthesis equals P left parenthesis x right parenthesis minus D left parenthesis x right parenthesis cross times q subscript m x to the power of m
    • Repeat steps 1 – 3 using the new polynomial R(x) in place of P(x) until the subtraction results in an expression for R(x) with degree less than the divisor
      • The quotient Q(x) is the sum of the terms you multiplied the divisor by: Q left parenthesis x right parenthesis equals q subscript m x to the power of m plus q subscript m minus 1 end subscript x to the power of m minus 1 end exponent plus... plus q subscript 1 x plus q subscript 0
      • The remainder R(x) is the polynomial after the final subtraction

Division by linear functions

  • If P(x) has degree n and is divided by a linear function (ax + b) then
    •  fraction numerator P open parentheses x close parentheses over denominator a x plus b end fraction equals Q open parentheses x close parentheses plus fraction numerator R over denominator a x plus b end fractionwhere 
      • ax + b is the divisor (degree 1)
      • Q(x) is the quotient (degree n – 1)
      • R is the remainder (degree 0)
    • Note that P left parenthesis x right parenthesis equals Q left parenthesis x right parenthesis cross times left parenthesis a x plus b right parenthesis plus R

Division by quadratic functions

  • If P(x) has degree n and is divided by a quadratic function (ax2 + bx + c) then
    •  fraction numerator P open parentheses x close parentheses over denominator a x squared plus b x plus c end fraction equals Q open parentheses x close parentheses plus fraction numerator e x plus f over denominator a x squared blank plus blank b x blank plus blank c end fraction where
      • ax2 + bx + c is the divisor (degree 2)
      • Q(x) is the quotient (degree n – 2)
      • ex + f is the remainder (degree less than 2)
    • The remainder will be linear (degree 1) if e ≠ 0, and constant (degree 0) if e = 0
    • Note that P left parenthesis x right parenthesis equals Q left parenthesis x right parenthesis cross times left parenthesis a x squared plus b x plus c right parenthesis plus e x plus f

Division by polynomials of degree k ≤ n

  • If P(x) has degree n and is divided by a polynomial D(x) with degree k ≤ n
    •  fraction numerator P open parentheses x close parentheses over denominator D left parenthesis x right parenthesis end fraction equals Q open parentheses x close parentheses plus fraction numerator R left parenthesis x right parenthesis over denominator D open parentheses x close parentheses end fraction where
      • D(x) is the divisor (degree k)
      • Q(x) is the quotient (degree n – k)
      • R(x) is the remainder (degree less than k)
    • Note that P left parenthesis x right parenthesis equals Q left parenthesis x right parenthesis cross times D left parenthesis x right parenthesis plus R left parenthesis x right parenthesis

Are there other methods for dividing polynomials?

  • Synthetic division is a faster and shorter way of setting out a division when dividing by a linear term of the form
    • To divide P left parenthesis x right parenthesis 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... plus a subscript 1 x plus a subscript 0 by left parenthesis x minus c right parenthesis:
      • Set b subscript n equals a subscript n
      • Calculate b subscript n minus 1 end subscript equals a subscript n minus 1 end subscript plus c cross times b subscript n
      • Continue this iterative process b subscript i minus 1 end subscript equals a subscript i minus 1 end subscript plus c cross times a subscript i
      • The quotient is Q left parenthesis x right parenthesis equals b subscript n x to the power of n minus 1 end exponent plus b subscript n minus 1 end subscript x to the power of n minus 2 end exponent plus... plus b subscript 2 x plus b subscript 1 and the remainder is r equals b subscript 0
  • You can also find quotients and remainders by comparing coefficients
    • Given a polynomial P left parenthesis x right parenthesis 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... plus a subscript 1 x plus a subscript 0
    • And a divisor D left parenthesis x right parenthesis equals d subscript k x to the power of k plus d subscript k minus 1 end subscript x to the power of k minus 1 end exponent plus... plus d subscript 1 x plus d subscript 0
    • Write Q left parenthesis x right parenthesis equals q subscript n minus k end subscript x to the power of n minus k end exponent plus... plus q subscript 1 x plus q subscript 0 and R left parenthesis x right parenthesis equals r subscript k minus 1 end subscript x to the power of k minus 1 end exponent plus... plus r subscript 1 x plus r subscript 0
    • Write P left parenthesis x right parenthesis equals Q left parenthesis x right parenthesis D left parenthesis x right parenthesis plus R left parenthesis x right parenthesis
      • Expand the right-hand side
      • Equate the coefficients
      • Solve to find the unknowns q’s & r’s

Exam Tip

  • In an exam you can use whichever method to divide polynomials - just make sure your method is written clearly so that if you make a mistake you can still get a mark for your method!

Worked Example

a)
Perform the division fraction numerator x to the power of 4 plus 11 x squared minus 1 over denominator x plus 3 end fraction. Hence write x to the power of 4 plus 11 x squared minus 1 in the form Q left parenthesis x right parenthesis cross times left parenthesis x plus 3 right parenthesis plus R.

2-7-2-ib-aa-hl-polynomial-division-a-we-solution-1-22-7-2-ib-aa-hl-polynomial-division-a-we-solution-2-2

b)
Find the quotient and remainder for fraction numerator x to the power of 4 plus 4 x cubed minus x plus 1 over denominator x squared minus 2 x end fraction. Hence write x to the power of 4 plus 4 x cubed minus x plus 1 in the form Q left parenthesis x right parenthesis cross times left parenthesis x squared minus 2 x right parenthesis plus R left parenthesis x right parenthesis.

2-7-2-ib-aa-hl-polynomial-division-b-we-solution-



  • 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
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 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
                                                                                                                                                                                                                                                                                                        Lucy Kirkham

                                                                                                                                                                                                                                                                                                        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.


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