Practical_8.pdf
Question 1 §
Find all complex solutions of the following symmetric system:
x3+y3=37x+y=3
:(x+y)3=x3+3x2y+3xy2+y3
(x+y)3=x3+y3+3xy(x+y)
−910=xyafter substituting in the system
xy=−910x+y=3
Solving this, x⋅(3−x)=−910
x2−3x−910=0
9x2−27x−10=0:x=1827±33
x=−31,310
∴x=−31,y=310ORx=310,y=−31
Question 2 §
Find a polynomial g(y) such that
g(1+x1)=x3+x31
(x+x1)3=x3+x31+3(x+x1)
g(1+x1)=(x+x1)3−3(x+x1)
∴g(z)=z3−3z:z=x+x1
Question 3 §
Let α and β be the complex roots of the polynomial x2+x+2. Without computing α and β separately, compute α2+β2, α3+β3 and α4+β4.
α+β=−1αβ=2
:(α+β)4=β4+4αβ3+8α2β2+4α3β+α4
1=(α4+β4)+4αβ(α2+β2+2αβ)
:(α+β)2−2αβ=α2+β2
1=(α4+β4)+4αβ(((α+β)2−2αβ)+2αβ)
α4+β4=1−4⋅2⋅(((−1)2−2⋅2)+2⋅2)
α4+β4=−7
α3+β3=5 from previous questions
α2+β2=−3
Question 4 §
Let α,β,γ be the complex roots of the polynomial 2x3−4x−6. Find a polynomial of degree three, with integer coefficients, having roots α1,β1,γ1.
q(x)=−6x3−4x2+2 by reading the coefficients in reverse
Alternative Method
q(x)=x3+p(x1)
q(x)=x3+(x32−x4−6)
q(x)=−6x3−4x2+2
Question 5 §
Let α,β,γ be the complex roots of the polynomial x3+2x2−x+3. Compute α2+β2+γ2 and α−1,β−1,γ−1.
α+β+γ=−2αβ+αγ+βγ=−1αβγ=−3
α2+β2+γ2=(α+β+γ)2−2(αβ+αγ+βγ)=6
α1+β1+γ1=αβγαβ+αγ+βγ=31