The 'minimal polynomial' of a number z is the f(x) of smallest degree with integer coefficients having z as a root.
For example the minimal polynomial for z = √3 is f(x) = x
2 - 3.
Determine the minimal polynomials of:
A) z = √5 + 3√2
B) z = tan18o.
To save typing square and cube roots, let s=sqrt(5) and c=cuberoot(2) and cc=cuberoot(4)=c^2
square both sides
x^2-2cx+cc=5
rearrange and factor
x^2-5=c(2x-c) *
cube both sides
(x^2-5)^3=2(8x^3-12x^2c+6xcc-2)
rearrange and factor
(x^2-5)^3-16x^2+4=-12c(2x-c)
substitute the *
(x^2-5)^3-16x^3+4=-12x(x^2-5)
So an unsimplified polynomial is
(x^2-5)^3-16x^3+4+12x(x^2-5)
Which in standard form is
x^6 - 15x^4 - 4x^3 + 75x^2 - 60x - 121
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Posted by Jer
on 2024-08-16 22:54:39 |
Part A
