Cubic polynomials II

Algebra Level 5

Suppose two polynomials of degree 3, $f(x), g(x)$ have three distinct positive integer roots each, and there is no common root between both polynomials. (In other words, the set of roots of these polynomials has 6 distinct elements.)

Also, $f(x)-g(x)=r$ for some real number $r$ for all real values of $x$ .

If $S(P(x))$ denotes the sum of absolute values of coefficients of a polynomial $P(x)$ , find the minimum possible value of $S(f(x))+S(g(x))$ .

This problem is part of the set ... and polynomials

The answer is 180.

1 solution

Omkar Kamat
Dec 20, 2014

For 2 polynomials to have the same difference for all real values, all coefficients must be the same apart from the constant term. The sum of the roots and the squares of the roots have to be the same. Since the polynomials have positive integer roots,we see that we can check which roots work. Since we are looking for the smallest , the roots are 1,5,6 and 2,3,7.

We can use vieta's formulas to see that the answer is 180

How do you define a solution as 'smallest'?

Joel Tan - 6 years, 5 months ago

We want the minimum possible value for the absolute values of the coefficients. By vieta's formulas, we see that the coefficients will be the smallest if the roots are the lowest that they can be.

This only works as the question asks for positive integer roots. If the roots were not positive, we wouldn't be able to apply this logic.

Omkar Kamat - 6 years, 5 months ago

There are 6 variables, and the minimum may not occur when all are minimised.

Joel Tan - 6 years, 5 months ago

@Joel Tan – That has to be the minimum as we are considering absolute values of the coefficients.

Omkar Kamat - 6 years, 5 months ago

@Omkar Kamat – That's an intuition and is not rigorously proven.

Joel Tan - 6 years, 5 months ago

Cubic polynomials II

The answer is 180.

1 solution

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