1. Roots of polynomials

2. FM Roots of polynomials: Quartics
KUS objectives
BAT Derive and use the relationships between the roots of quartic equations
Starter: Solve
𝑥𝑥𝑥given that one root is 𝑧=1−𝑖
𝑥𝑥𝑥
𝑥=𝑥𝑥𝑥=−2±𝑖

3. 𝑎 𝑥 4 +𝑏 𝑥 3 +𝑐 𝑥 2 +𝑑𝑥+𝑒=0 has up to four roots .
Notes
𝑎 𝑥 4 +𝑏 𝑥 3 +𝑐 𝑥 2 +𝑑𝑥+𝑒=0 has up to four roots .
𝒂 𝑥 4 +𝒃 𝑥 3 +𝒄 𝑥 2 +𝒅𝑥+𝒆=𝑎 𝑥−𝛼 𝑥−𝛽 𝑥−𝛾 𝑥−𝛿
𝛼, 𝛽, 𝛾,𝛿 are the roots =…
=𝑎 𝑥 4 −𝑎 𝛼+𝛽+𝛾+𝛿 𝑥 3 +𝑎 𝜶𝜷+𝜶𝜸+𝜶𝜹+𝜷𝜸+𝜷𝜹+𝜸𝜹 𝑥 2
𝑎 𝛼𝛽𝛾+𝛼𝛽𝛿+𝛼𝛾𝛿+𝛽𝛾𝛿 𝑥+𝑎𝛼𝛽𝛾𝛿
In shorthand
𝛼= − 𝑏 𝑎
And 𝛼𝛽= 𝑐 𝑎
And 𝛼𝛽𝛾 =− 𝑑 𝑎
Equate the coefficients 𝛼+𝛽+𝛾+𝛿 =− 𝑏 𝑎
And 𝛼𝛽+𝛼𝛾+𝛼𝛿+𝛽𝛾+𝛽𝛿+𝛾𝛿 = 𝑐 𝑎
And 𝛼𝛽𝛾+𝛼𝛽𝛿+𝛼𝛾𝛿+𝛽𝛾𝛿 =− 𝑑 𝑎
And 𝛼𝛽𝛾𝛿 = 𝑒 𝑎

4. WB C1a The quartic equation 𝑥 4 +2 𝑥 3 +𝑝 𝑥 2 +𝑞𝑥−60=0 has roots ∝,𝛽, 𝛾, 𝛿
Given that 𝛾=−2+4𝑖 and δ= 𝛾 ∗
a) show that ∝+𝛽−2= and that 𝛼𝛽+3=0
Hence find all the roots of the quartic and the values of p and q
𝛼+𝛽+𝛾+𝛿 =− 𝑏 𝑎
Gives 𝛼+𝛽+ −2+4𝑖 + −2−4𝑖 =−2
Gives 𝛼+𝛽−2=0 QED (1)
𝛼𝛽𝛾𝛿= 𝑒 𝑎
Gives 𝛼𝛽 −2+4𝑖 −2−4𝑖 =− (2)
Substitute (1) into (2) Gives 𝛼(2−𝛼) −2+4𝑖 −2−4𝑖 =−60
𝛼 2−𝛼 (20)=−60
𝛼 2 −2𝛼−3=0
(𝛼−3)(𝛼+1)=0 so 𝛼=3, −1
𝛽=−1, 3
So the roots are 3, −1, −2±4𝑖

5. Given that 𝛾=−2+4𝑖 and δ= 𝛾 ∗ a) show that ∝+𝛽−2=0 and that 𝛼𝛽+3=0
WB C1b The quartic equation 𝑥 4 +2 𝑥 3 +𝒑 𝑥 2 +𝒒𝑥−60=0 has roots ∝,𝛽, 𝛾, 𝛿
Given that 𝛾=−2+4𝑖 and δ= 𝛾 ∗
a) show that ∝+𝛽−2= and that 𝛼𝛽+3=0
b) Hence find all the roots of the quartic and the values of p and q
In shorthand
𝛼= − 𝑏 𝑎
And 𝛼𝛽= 𝑐 𝑎
And 𝛼𝛽𝛾 =− 𝑑 𝑎
So the roots are 3, −1, −2+4𝑖, −2−4𝑖
So 𝛼𝛽+𝛼𝛾+𝛼𝛿+𝛽𝛾+𝛽𝛿+𝛾𝛿 = 𝑐 𝑎
Gives 𝑝= −3 + −6+12𝑖 + −6−12𝑖 + 2−4𝑖 + 2+4𝑖 + 20
𝑝=9
So 𝛼𝛽𝛾+𝛼𝛽𝛿+𝛼𝛾𝛿+𝛽𝛾𝛿 =− 𝑑 𝑎
gives −𝑞= 6−12𝑖 𝑖 −20 =52
So 𝑞=−52

6. WB C The d
NOW DO Ex 4C

7. self-assess
One thing learned is –
One thing to improve is –

8. END