1. Roots of polynomials

2. FM Roots of polynomials: Related Expressions
KUS objectives
BAT Evaluate expressions related to the roots of polynomials
Starter: Solve
𝑥𝑥𝑥given that one root is 𝑧=1−𝑖
𝑥𝑥𝑥
𝑥=𝑥𝑥𝑥=−2±𝑖

3. Notes Spot the patterns I
1 𝛼 + 1 𝛽 =
𝛼+𝛽 𝛼𝛽
Rules for reciprocals
1 𝛼 + 1 𝛽 + 1 𝛾 =
𝛼𝛽+𝛼𝛾+𝛽𝛾 𝛼𝛽𝛾
1 𝛼 + 1 𝛽 + 1 𝛾 + 1 𝛿 =
𝛼𝛽𝛾+𝛼𝛽𝛿+𝛼𝛾𝛿+𝛽𝛾𝛿 𝛼𝛽𝛾𝛿
Rules for powers
𝛼 𝑛 × 𝛽 𝑛 =
𝛼𝛽 𝑛
𝛼 𝑛 × 𝛽 𝑛 × 𝛾 𝑛 =
𝛼𝛽𝛾 𝑛
𝛼 𝑛 × 𝛽 𝑛 × 𝛾 𝑛 × 𝛿 𝑛 =
𝛼𝛽𝛾𝛿 𝑛

4. Rules for roots of polynomials used earlier in this topic
Notes Spot the patterns II
Rules for roots of polynomials used earlier in this topic
𝛼 2 × 𝛽 2 = 𝛼+𝛽 2 −2𝛼𝛽
𝛼 3 + 𝛽 3 = 𝛼+𝛽 3 −3𝛼𝛽 𝛼+𝛽
There are equivalent results for higher powers
We can use these to find expressions for sums of squares and sums of cubes

5. 𝑎) 𝛼+𝛽+𝛾 2 = 𝛼 2 +𝛼𝛽+𝛼𝛾+𝛽𝛼+ 𝛽 2 +𝛽𝛾+𝛾𝛼+𝛾𝛽+ 𝛾 2
WB D1 Sums of squares
Expand 𝛼+𝛽+𝛾 2
A cubic equation has roots 𝛼, 𝛽, 𝛾 such that 𝛼𝛽+𝛽𝛾+𝛼𝛾=7 and 𝛼+𝛽+𝛾=−3 Find the value of 𝛼 2 + 𝛽 2 + 𝛾 2
𝑎) 𝛼+𝛽+𝛾 2 = 𝛼 2 +𝛼𝛽+𝛼𝛾+𝛽𝛼+ 𝛽 2 +𝛽𝛾+𝛾𝛼+𝛾𝛽+ 𝛾 2
𝛼+𝛽+𝛾 2 = 𝛼 2 + 𝛽 2 + 𝛾 2 +2 𝛼𝛽+𝛽𝛾+𝛼𝛾
𝑏) 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑔𝑖𝑣𝑒𝑠 −3 2 = 𝛼 2 + 𝛽 2 + 𝛾 2 +2(7)
𝛼 2 + 𝛽 2 + 𝛾 2 =−5

6. Notes Spot the patterns III
Rules for sums of squares
Quadratic 𝛼 2 + 𝛽 2 =
𝛼+𝛽 2 −2 𝛼𝛽
Cubic 𝛼 2 + 𝛽 2 + 𝛾 2 =
𝛼+𝛽+𝛾 2 −2 𝛼𝛽+𝛽𝛾+𝛼𝛾
Quartic 𝛼 2 + 𝛽 2 + 𝛾 2 + 𝛿 2
= 𝛼+𝛽+𝛾+𝛿 2 −2 𝛼𝛽+𝛼𝛾+𝛼𝛿+𝛽𝛾+𝛽𝛿+𝛾𝛿
Rules for sums of cubes
Quadratic 𝛼 3 + 𝛽 3 =
𝛼+𝛽 3 −3𝛼𝛽 𝛼+𝛽
Cubic 𝛼 3 + 𝛽 3 + 𝛾 3
= 𝛼+𝛽+𝛾 3 −3 𝛼+𝛽+𝛾 𝛼𝛽+𝛽𝛾+𝛼𝛾 +3𝛼𝛽𝛾

7. NOW DO Ex 4D WB D2 The three roots of a cubic equation are 𝛼, 𝛽 ,𝛾
Given that 𝛼𝛽𝛾=4; 𝛼𝛽+𝛽𝛾+𝛼𝛾=−5 and 𝛼+𝛽+𝛾=3
Find the value of (𝛼+3)(𝛽+3)(𝛾+3)
𝛼+3 𝛽+3 𝛾+3 =
=𝛼𝛽𝛾+3𝛼𝛽+3𝛼𝛾+9𝛼+3𝛽𝛾+9𝛽+9𝛾+27
=𝛼𝛽𝛾+3 𝛼𝛽+𝛼𝛾+𝛽𝛾 +9(𝛼+𝛽+𝛾)+27
=4 +3 −5 +9(3)+27 = 43
NOW DO Ex 4D

8. One thing to improve is –
KUS objectives
BAT Evaluate expressions related to the roots of polynomials
self-assess
One thing learned is –
One thing to improve is –

9. END

10. WB 6 The region R is bounded by the curve 𝑥= 2𝑦−1 , the y-axis and the vertical lines y=4 and y = 8
Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π
𝑥= 2𝑦−1
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦− 𝑑𝑦
𝑣𝑜𝑙𝑢𝑚𝑒=𝜋 𝑦−1 𝑑𝑦
= 𝜋 𝑦 2 −𝑦 8 4
= 𝜋 64−8 −𝜋 16−4 =44𝜋