What is the smallest degree monic polynomial with a/b as a root?

  1. Let a and b be the two complex roots of the polynomial 2x³+2x+1. Find, with proof, the smallest degree monic polynomial with integer coefficients with a/b as a root.

    Answer by Josh Swanson
    Since you haven’t gotten an answer yet, I’ll mention that Wolfram Alpha can at least compute the minimal polynomial in question: x^6 + 3x^5 + 10x^4 + 15x^3 + 10x^2 + 3x + 1. Naively plugging in awful expressions for the roots in would prove that the ratio is a root, and there are deterministic algorithms for proving the irreducibility of polynomials over Q, but this is a terrible and backwards approach.

  2. I was put in AP Physics B for next year because of my high math grades in middle school. Also, I have a strong interest in math and science. I am wondering how much work I will need to put towards this.

    Answer by Imre
    You don’t need to put real work into it. You like it, so it’s mainly fun rather than work. Have fun acquiring some interesting, nice knowledge!

  3. I play the B♭ Clarinet, and i bought a box of all bad reeds except for 2 of them!! Is there a way that i can make them good again? I bought the exact brand (Rico) and size (3) that i usually use, but one 2 of them were good! Also, how do i make it easier to switch octaves? I have a hard time going from A to B.

    Answer by Just Me
    For starters, don’t buy Rico again. They’re cheap and inconsistent as you have probably already figured out. Second, there are many ways to modify your reeds. I always begin by checking to see if they are warped. Check this by placing it on a flat surface and gently pressing on either side as if to rock it back and forth. If it rocks, it’s warped and you need to sand down the flat side until it stays put. Also, if you have a reed knife, you can slightly shave away at the wings. This will help with the resistance. Work on that part slowly and keep testing it in between shavings so you don’t overdo it. Good luck!

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