Prior content: Complex Numbers
Previous Section: Argand Diagrams
Next Section: Series
Contents
- Whole Topic Notes
- 4A Roots of Quadratics
- 4B Roots of Cubics
- 4C Roots of Quartics
- 4D Expressions Relating to Roots of Polynomials
- 4E Linear Transformations of Roots
- Whole Topic Summary Resources (Including Past Paper Questions)
Whole Topic Notes
Download Full Topic Slides
- Textbook Notes as PPTs (Courtesy of Owen134866)
Full Notes to Accompany Slides
Dr Frost
- Dr Frost AS Core Pure Roots of Polynomials Section
- Dr Frost AS Core Pure Roots of Polynomials Slides
- Blank Notes to Follow Slides Word Doc or PDF (Courtesy of King’s Chester Maths Department)
Acute Learning Whole Chapter Problem Pairs
Notes
- Physics & Maths Tutor
- CP1 Ch.4 Roots of Polynomials: Cheat sheet
4A Roots of Quadratics
Textbook resources
Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.
Dr Frost resources
Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths
Download Blank Notes
(Courtesy of King’s Chester Maths Department)
Download Slides
Extra Questions
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 675a: Determine the sum or product of roots of a quadratic equation. (not yet active)
- 675b: Determine the sum of reciprocals of roots of a quadratic equation. (not yet active)
- 675c: Given the roots of a quadratic equation, determine integer values for its coefficients. (not yet active)
- 675: Exam Practice: Relationship between roots and coefficients of polynomial equations, up to quartics
Other resources
3Blue1Brown Linear transformations and matrices | Chapter 3, Essence of linear algebra
Alternative Handouts
- Acute Learning Problem Pairs as PowerPoint or PDF
Online Platform
- Bossmaths interactive 2×2 transformations and questions
Worksheets
- MathedUp
- MadAsMaths (T. Madas)
- Roots of Polynomial Equations Quadratics Q1–19 (difficulty 3–5 of 5)
- Pret Homework (@MathedUp)
- Roots of Quadratic Equations (@MathedUp, pptx)
4B Roots of Cubics
Textbook resources
Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.
Dr Frost resources
Download Blank Notes
(Courtesy of King’s Chester Maths Department)
Download Slides
Extra Questions
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 675d: Determine the sum or roots, sum or root products or product of all roots of a cubic equation. (not yet active)
- 675e: Determine, using formulae, the coefficients of a cubic equation given its roots, where two are complex. (not yet active)
- 675g: Determine the sum of reciprocals of roots of a cubic equation. (not yet active)
- 675m: Determine roots of a cubic with some unknown coefficients, where expressions for the roots are given in terms of a single constant. (not yet active)
- 675n: Determine two unknown coefficients of a cubic equation given one real root and an area formed by all the roots on an Argand diagram. (not yet active)
- 675: Exam Practice: Relationship between roots and coefficients of polynomial equations, up to quartics
Other resources
Enrichment Video
Mathologer: 500 years of not teaching the cubic formula
Alternative Handouts
- Acute Learning Problem Pairs as PowerPoint or PDF
Online Platform
- Bossmaths interactive 2×2 transformations and questions
Worksheets
- MadAsMaths (T. Madas)
- Polynomials Q13 (difficulty 4 of 5)
- Roots of Polynomial Equations Cubics Q2, 7, 12–14, 21, 23–27, 29, 31, 33, 39 (difficulty 2–5 of 5)
4C Roots of Quartics
Textbook resources
Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.
Dr Frost resources
Download Blank Notes
(Courtesy of King’s Chester Maths Department)
Download Slides
Extra Questions
- Dr Frost Questions (very short and basic)
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
Other resources
Alternative Handouts
- Acute Learning Problem Pairs as PowerPoint or PDF
Online Platform
- Bossmaths interactive 3×3 transformations and questions
4D Expressions Relating to Roots of Polynomials
Textbook resources
Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.
Dr Frost resources
Download Blank Notes
(Courtesy of King’s Chester Maths Department)
Download Slides
Extra Questions
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 675h: Determine the value of an expression (alpha + k)(beta + k)(gamma + k) for some constant k (not yet active)
- 675i: Determine the sum of squares of roots alpha2 + beta2 + gamma2 of a cubic equation by use of an appropriate identity. (not yet active)
- 675j: Determine the sum of cubes of roots alpha3 + beta3 + gamma3 by use of an appropriate identity. (not yet active)
- 675k: Determine the sum alpha4 + beta4 by use of an appropriate identity. (not yet active)
- 675l: Use identities for (alpha ± beta)2 to determine values for expressions involving roots of a polynomial alpha and beta (not yet active)
- 675: Exam Practice: Relationship between roots and coefficients of polynomial equations, up to quartics
Other resources
Alternative Handouts
- Acute Learning Problem Pairs as PowerPoint or PDF
Online Platform
- Bossmaths interactive 2×2 transformations and questions
Worksheet & Activities
- MEI activities from old FP1
Worksheets
- MadAsMaths (T. Madas)
- Equations: Advanced Skills Q96 (difficulty 5 of 5)
- Roots of Polynomial Equations Cubics Q1, 4, 8, 10, 16, 28, 30, 32, 34, 35, 40 (difficulty 2–5 of 5)
4E Linear Transformations of Roots
Textbook resources
Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.
Dr Frost resources
Download Blank Notes
(Courtesy of King’s Chester Maths Department)
Download Slides
Extra Questions
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 676a: Determine a quadratic equation whose roots are a linear transformation kalpha + c of the original roots. (not yet active)
- 676b: Determine a cubic or quartic equation whose roots are a linear transformation kalpha + c of the original roots. (not yet active)
- 676c: Determine a quadratic equation whose roots are the reciprocals of the roots of another quadratic equation. (not yet active)
- 676d: Determine a cubic equation whose roots are the reciprocals of the roots of another cubic equation. (not yet active)
- 676e: Determine a quadratic equation whose roots are the squares of the roots of another quadratic equation. (not yet active)
- 676f: Determine a quadratic equation whose new roots are a mixture of alpha and beta (not yet active)
- 676: Exam Practice: Polynomial equations whose roots are a linear transformation of the roots of a given polynomial equation
Other resources
Alternative Handouts
- Acute Learning Problem Pairs as PowerPoint or PDF
Worksheets
- MadAsMaths (T. Madas)
- Transformations Of Graphs Exam Questions Q36 (difficulty 3 of 5)
- Roots of Polynomial Equations Cubics Q3, 5, 6, 9, 11, 17–20, 22, 36, 38 (difficulty 2–4 of 5)
Whole Topic Summary Resources (Including Past Paper Questions)
Activities
- Andy Lutwyche
Download PPQs as WW Powerpoint for Using Roots of Polynomials
External whole topic past paper style questions
- WW Dr Frost Style QQQs (Version 1) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
- WW Dr Frost Style QQQs (Version 2) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
- Advanced Highers Matrices Exam Questions & Answers
Worksheets
- Mathshelper (Jon Stone)
- MathedUp
- Jethwa Maths
- 10 Minute Starters: Roots and Polynomials
- Physics & Maths Tutor
- CP1 Ch.4 Roots of Polynomials: Questions by topic
- Dr Austin Maths