## Finding of a polynomial

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: $$f(x)=ax^3-9ax^2+26ax-18a$$.