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March 25, 2017

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A number is called algebraic if there is a polynomial with rational coefficients for which the number is a root. For example,√2 is algebraic because it is a root of the polynomial x^2−2. The number √2+√3+√5 is also algebraic because it is a root of a monic polynomial of degree 8, namely x^8+ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h. Find |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h|.

  • Math (Algebra) - ,

    If the coefficients are all rational, then the irrational roots must occur in conjugate pairs. So, the polynomial is

    (x-(√2+√3+√5))
    (x-(√2+√3-√5))
    (x-(√2-√3+√5))
    (x-(√2-√3-√5))
    (x-(-√2+√3+√5))
    (x-(-√2+√3-√5))
    (x-(-√2-√3+√5))
    (x-(-√2-√3-√5))

    x^8 - 40x^6 + 352x^4 - 960x^2 + 576

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