Finding of a polynomial
Task number: 2634
Find a cubic real polynomial \(f\) with integer coefficients satisfying \((x-1)|f\) and also \(f(2)=f(3)=f(4)\).
Derive relationships between the coefficients of \(f(x)=ax^3+bx^2+cx+d\).
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.
The exercise has more solutions, e.g. \(f(x)=-x^3+9x^2-26x+18\).
The general formula is: \(f(x)=ax^3-9ax^2+26ax-18a\).