Questions asked by nicolina

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• Prove that the number #(sqrt(2)+sqrt(17)-sqrt(19))# is irrational?
  
• If #n# is a natural number, prove that the number #(sqrt(n) + sqrt(n+1))# is irrational?
  
• Find all rational numbers, so that the number #(a+sqrt(3))(4-a-sqrt(3))# is rational?
  
• How do you factor #a^3-2ab^2-b^3#?
  
• If numbers α and β are irrational, and α+β rational, prove that the numbers α-β and α+2β are irrational?
  
• How do you factor #x^8+2x^6+3x^4+2x^2+1#?
  
• How do you factor #x^4+2x^3y-3x^2y^2-4xy^3-y^4#?
  
• How do you factor #x^4+2x^3y−3x^2y^2−4xy^3−y^4# without quadratic equation?
  
• Prove that polynomial p(x)=ax^2+bx+c with integer coefficients that do not simultaneously equal zero does not exist, such that p(2^(1/3))=0?
  
• If #a+b+c=0#, prove that #(a^2+b^2+c^2)^2=2(a^4+b^4+c^4)#?
  
• If #x+y+z=1#, #x^2+y^2+z^2=1# and #x^3+y^3+z^3=1#, prove that #xyz=0#?
  
• Determine the numbers a,b and c so that the polynomials #x+5# and #a(x-2)(x-3)+b(x-1)(x-3)+c(x-1)(x-2)# are identical?
  
• For which natural numbers #m# is polynomial #(x+1)^m+(x-1)^m# divisible by x?
  
• Do #x,y,z,t ∈ Q# exist so that #(x+y√2)^2+(z+t√2)^2 = 5+4√2#?
  
• Prove that the number #sqrt(1+sqrt(2+...+sqrt(n)))# is not rational for any natural number #n# greater than #1#?
  
• If #x+1/x=1#, what is the value of #x^2001+1/x^2001# ?
  
• If #f(x/(x+1))=(x-1)^2#, what is #f(3)#?
  
• Solve the inequality #-1/2<x^3/(1+x^6)#?
  
• Let #a,b,c# be positive real numbers such that #a+b+c=1#.Prove the following inequality: #(a+1/a)^2+(b+1/b)^2+(c+1/c)^2>=100/3#?
  
• If #x, y, z# are positive real numbers such that #x+y+z = 1#. Prove the following inequality: #xy + yz + zx − xyz ≤ 8/27#?
  
• If #z+1/z=1#, find #z^3#, and then #z^1000+1/z^1000#?