perplexus.info :: Just Math : Two Mini-Pollies

Two Mini-Pollies (Posted on 2024-08-15)

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) = x² - 3.

Determine the minimal polynomials of:

A) z = √5 + ³√2
B) z = tan18^o.

No Solution Yet Submitted by K Sengupta

Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

Part B by looking up z

| Comment 2 of 3 |

It's possible to do this using trig identities

z=tan(pi/10)=tan((1/2)(pi/5) etc,

but I just looked it up: z = (1/5)sqrt(25-10sqrt(5))

x = (1/5)sqrt(25-10sqrt(5))

5x = sqrt(25-10sqrt(5))

25x^2 = 25 - 10sqrt(5)

25(x^2-1) = -10sqrt(5)

625(x^4-2x^2+1) = 500

5(x^4-2x^2+1)=4

5x^4-10x^2+1=0

Posted by Jer on 2024-08-16 23:08:57

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