A-level Further Maths Year 12 · Topic 3
Algebra, series and proof
This topic links the roots of a polynomial to its coefficients.
We will transform equations, sum series using standard results and the method of differences.
Then we prove results for every whole number using proof by induction.
- 5 subtopics
- 135 questions
- Free: 3.1 notes and 10 questions
In this topic
- 3.1Roots of polynomials: relationships between roots and coefficientsFree below · 10 questions
- 3.2Transforming equations: new roots by substitutionOpen in Full Marks · 21 questions
- 3.3Series: sigma notation and the standard results for Σr, Σr² and Σr³Open in Full Marks · 50 questions
- 3.4The method of differences and partial fractionsOpen in Full Marks · 27 questions
- 3.5Proof by induction: series, divisibility and matricesOpen in Full Marks · 27 questions
3.1 · Free notes
Roots of polynomials: relationships between roots and coefficients
Key points
- For ax² + bx + c = 0 with roots α, β: α + β = −b/a and αβ = c/a.
- For ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, αβγ = −d/a.
- For a quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
- Useful identities: α² + β² = (α + β)² − 2αβ; 1/α + 1/β = (α + β)/αβ; α² + β² + γ² = (Σα)² − 2Σαβ.
- To write an equation with given roots: x² − (sum)x + (product) = 0.
In more detail
- Worked example: roots of x² − 5x + 3 = 0 are α, β. An equation with roots α², β² has sum α² + β² = 25 − 6 = 19 and product (αβ)² = 9: x² − 19x + 9 = 0.
- Sign pattern: for a cubic the signs alternate: −b/a, +c/a, −d/a.
- Exam tip: write the sums and products down before manipulating; never try to find the roots themselves unless asked.
Practice
Roots of polynomials: relationships between roots and coefficients: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. For 2x² − 6x + 5 = 0, what is α + β?
- −3
- 3
- 5/2
- 6
Show answer
B: 3. −b/a = 6/2.
2. For 2x² − 6x + 5 = 0, what is αβ?
- −5/2
- 3
- 5
- 5/2
Show answer
D: 5/2. c/a.
3. If α + β = 4 and αβ = 1, what is α² + β²?
- 16
- 18
- 14
- 15
Show answer
C: 14. 16 − 2 = 14.
4. For x³ − 2x² + 3x − 7 = 0, what is αβγ?
- 7
- −7
- 3
- 2
Show answer
A: 7. −d/a = 7.
5. For x³ − 2x² + 3x − 7 = 0, what is Σαβ?
- −3
- 3
- 2
- 7
Show answer
B: 3. c/a.
6. Write a quadratic with roots 2 and −5.
- x² − 3x − 10 = 0
- x² + 3x + 10 = 0
- x² − 7x + 10 = 0
- x² + 3x − 10 = 0
Show answer
D: x² + 3x − 10 = 0. Sum −3, product −10.
7. If α + β = 3 and αβ = 2, what is 1/α + 1/β?
- 2/3
- 5
- 3/2
- 1
Show answer
C: 3/2. (α + β)/αβ.
8. For x³ + 4x² − x + 6 = 0, what is α + β + γ?
- −4
- 4
- −6
- 1
Show answer
A: −4. −b/a.
9. If Σα = 2 and Σαβ = −3 for a cubic, what is α² + β² + γ²?
- 4
- 10
- −2
- 7
Show answer
B: 10. 4 − 2(−3) = 10.
10. For x⁴ − 3x³ + x − 5 = 0, what is αβγδ?
- 5
- 3
- 1
- −5
Show answer
D: −5. e/a.
Keep going in Full Marks
Full Marks has 10 questions on Roots of polynomials: relationships between roots and coefficients, set out in practice levels. You also get notes and 3D models for all 5 subtopics of Algebra, series and proof, plus games, timed exams and past papers.
On your exam board
- AQATopic 3: Further algebra and functions
- OCR ATopic 1: Proof, Topic 5: Roots of polynomials and series
- OCR BTopic 4: Roots of polynomials, Topic 5: Sequences, series and induction
- EdexcelTopic 2: Series, Topic 3: Roots of polynomials, Topic 6: Proof by induction
- CambridgeTopic 1: Roots of polynomial equations, Topic 3: Summation of series, Topic 7: Proof by induction