Maths Extension 1 Year 11 - Module 2 - Lesson 9
Roots and Coefficients, General Polynomials
Name
Date
Class
1. Lesson focus
General polynomial roots and coefficients. Vieta's relations connect coefficients to elementary symmetric sums; signs alternate in the monic polynomial expansion.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. For 2x³−6x²−4x+8=0, find the sum and product of roots.
3. Form the monic cubic whose roots are 1,2 and −3.
3. Challenge
A monic quartic has roots α,β,γ,δ. State the coefficient of x³ in terms of their sum.
4. Multiple choice
For ax³+bx²+cx+d, the sum of roots is what?
A. −b/a
B. b/a
C. −c/a
D. d/a
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. For x³+2x²−5x−6=0, state the sum, pairwise-product sum and product of roots.
Sum=−2, pairwise sum=−5, product=6.
2. For 2x³−6x²−4x+8=0, find the sum and product of roots.
Sum=−(−6)/2=3; product=−8/2=−4.
3. Form the monic cubic whose roots are 1,2 and −3.
Sum=0; pairwise sum=2−3−6=−7; product=−6. Polynomial x³−7x+6.
Challenge. A monic quartic has roots α,β,γ,δ. State the coefficient of x³ in terms of their sum.
The coefficient is −(α+β+γ+δ), by expanding ∏(x−root).
Multiple choice. For ax³+bx²+cx+d, the sum of roots is what?
Correct answer: A. The x² coefficient is −a times the sum of the roots.
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