Maths Extension 1 Year 11 - Module 2 - Lesson 7
Sums and Products of Roots, Cubics
Name
Date
Class
1. Lesson focus
Sums and products of cubic roots. For ax³+bx²+cx+d=0, the sum of roots is −b/a, pairwise-product sum c/a and product −d/a.
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. The roots are 1,2,3. Find the sum of their squares.
3. For roots α,β,γ with sum 4 and pairwise-product sum 3, find αβγ if their monic cubic is x³−4x²+3x+8.
3. Challenge
For x³+2x²−5x−6=0, state the sum and product of roots.
4. Multiple choice
For x³−6x²+11x−6, the product of its roots is what?
A. 11
B. −6
C. 6
D. −11
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. State the sum, pairwise-product sum and product of roots of x³−6x²+11x−6.
Sum=6, pairwise sum=11, product=6.
2. The roots are 1,2,3. Find the sum of their squares.
1²+2²+3²=14; also (1+2+3)²−2(1·2+1·3+2·3)=36−22=14.
3. For roots α,β,γ with sum 4 and pairwise-product sum 3, find αβγ if their monic cubic is x³−4x²+3x+8.
Vieta gives product=−8, since the constant term is −αβγ=8.
Challenge. For x³+2x²−5x−6=0, state the sum and product of roots.
Sum=−2; product=6.
Multiple choice. For x³−6x²+11x−6, the product of its roots is what?
Correct answer: C. For a monic cubic the product is the negative of the constant term: −(−6)=6.
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