Re: HSC 2016 4U Marathon
b) The polynomial equationx4+ 4x3−2x2−12x−3 = 0 has rootsα,β,γandδ.Find the polynomial equation with roots (α+ 1), (β+ 1), (γ+ 1) and (δ+ 1).(c) Hence, or otherwise, solve the equationx4+ 4x3−2x2−12x−3 = 0.
Use _^_ notation to denote a power.
Replace x with x-1 for the equation with roots (α+ 1) etc.
(x-1)^4 + 4(x-1)^3 - 2(x-1)^2 - 12(x-1) - 3 = 0
With the help of the binomial theorem this is simplified to
x^4-8x^2+4=0
Use completing the square to factorise said expression:
x^4-8x^2+16=12
(x^2-4)^2=12
x^2-4=+-2sqrt(3)
x^2=4+-2sqrt(3)
Hence:
x=sqrt(4+2sqrt3)
x=sqrt(4-2sqrt3)
x=-sqrt(4+2sqrt3)
x=-sqrt(4-2sqrt3)
If one can see ahead and identify that:
(1+sqrt3)^2 = 1 + 2sqrt3 + 3 = 4 + 2sqrt(3)
and similarly (1-sqrt3)^2 = 4 - 2sqrt(3)
The roots can be rewritten as:
x=1+sqrt3
x=1-sqrt3
x=-1+sqrt3
x=-1-sqrt3
Now, note that these are the roots (α+ 1) etc. because we already established these are the roots to the polynomial equation which we just solved.
Hence:
α=sqrt3
β=-sqrt3
γ=-2+sqrt3
δ=-2-sqrt3