Finding of a polynomial

Task number: 2634

Find a cubic real polynomial \(f\) with integer coefficients satisfying \((x-1)|f\) and also \(f(2)=f(3)=f(4)\).

  • Hint

    Derive relationships between the coefficients of \(f(x)=ax^3+bx^2+cx+d\).

  • Resolution

    From \((x-1)|f\) follows that \(x=1\) is a root of the polynomial, i.e. \(a+b+c+d=0\).

    By the substitution for \(x\) we get also equations \(8a+4b+2c+d=27a+9b+3c+d=64a+16b+4c+d\).

    By a suitable subtraction we get \(19a+5b+c=0\) and \(37a+7b+c=0\), and consequently \(9a+b=0\). For an integer \(a\), e.g. \(a=-1\) we may calculate the remaining coefficients.

  • Result

    The exercise has more solutions, e.g. \(f(x)=-x^3+9x^2-26x+18\).

    The general formula is: \(f(x)=ax^3-9ax^2+26ax-18a\).

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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