Anything's possible, am I right?

The polynomials x2+2x^2 + 2 and x2+x1x^2+x-1 are written on a board. If the polynomials f(x)f(x) and g(x)g(x) are already on the board (f(x)f(x) may equal to g(x)g(x)) we are allowed to add any of f(x)g(x),f(x)g(x)f(x) g(x), f(x) -g(x) or f(x)+g(x)f(x) + g(x) to the board.

(a) Can the polynomial xx ever be made to appear on the board?

(b) Can the polynomial 4x14x-1 ever be made to appear on the board?

Give proof.


  • f(x)f(x) and g(x)g(x) can represent any two polynomials on the board

Note by Sharky Kesa
3 years, 11 months ago

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(a) Suppose a,ba, b are integers. If f(a),g(a)f(a), g(a) are divisible by bb, then f(a)+g(a),f(a)g(a),f(a)g(a)f(a)+g(a), f(a)-g(a), f(a)g(a) are all divisible by bb as well. Thus the property "divisible by bb at x=ax = a" is an invariant; they will hold for all polynomials generated.* At x=3x = 3, f(x)=g(x)=11f(x) = g(x) = 11 and thus both of them are divisible by 1111, so all polynomials generated will be divisible by 1111 on x=3x=3. But xx doesn't satisfy this (since it gives 33 on x=3x=3). So it cannot appear.

(*) We can use induction to formalize this, inducting on the number of operations required to generate a polynomial. The base case is f,gf, g that takes zero steps; the induction step lies on the fact that if a polynomial pp can be generated (in the shortest way) from either of f+g,fg,fgf+g, f-g, fg, then f,gf,g must be generated before pp, so we can use induction there.

(b) It can appear. Let F1(x)=x2+2,F2(x)=x2+x1F_1(x) = x^2 + 2, F_2(x) = x^2 + x - 1. Then,

  • F3(x)=F1(x)F2(x)=x3F_3(x) = F_1(x) - F_2(x) = x - 3
  • F4(x)=F3(x)F3(x)=x26x+9F_4(x) = F_3(x) F_3(x) = x^2 - 6x + 9
  • F5(x)=F1(x)F4(x)=7x10F_5(x) = F_1(x) - F_4(x) = 7x - 10
  • F6(x)=((F5(x)F3(x))F3(x))F3(x)=4x1F_6(x) = ((F_5(x) - F_3(x)) - F_3(x)) - F_3(x) = 4x - 1

Ivan Koswara - 3 years, 11 months ago

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What is the reason for choosing x=3 x = 3 ? In particular, if we were given 2 other (quadratic) polynomials, how do we know what to use?

Can we classify the set of polynomials which can be reached through these operations?

Hint: Bezout's Identity.

Calvin Lin Staff - 3 years, 11 months ago

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Nice question :)

Hint: Find an Invariant.

Hint: Evaluate the polynomial functions at a particular point.

Calvin Lin Staff - 3 years, 11 months ago

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I don't really get your problem. What are g(x) and f(x)? Are g(x) and f(x) equal to x2+1x^{2}+1 and/or x2+x1x^{2} +x-1

Julian Poon - 3 years, 11 months ago

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I think f(x) and g(x) some other polynomial function

Department 8 - 3 years, 11 months ago

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It's any f(x),g(x)f(x), g(x) that is written on the board.

Jake Lai - 3 years, 11 months ago

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f(x)f(x) and g(x)g(x) are any two polynomials on the board.

Sharky Kesa - 3 years, 11 months ago

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