Polynomials roots

[narinder]

Bruce loves learning about polynomials. One day, Bruce found a set of polynomials in the form ax²+bx+c that met the following conditions: 1) All roots are prime. 2) The sum of the roots of each polynomial is 24. 3) a always equals 1. Find the sum of the c values of all polynomials in Bruce's set. This polynomial is true for (-3,27),(-7,31)... I don't know how to continue and find sum of c values of all polynomials.

 

[Ishuda]

Bruce loves learning about polynomials. One day, Bruce found a set of polynomials in the form ax²+bx+c that met the following conditions: 1) All roots are prime. 2) The sum of the roots of each polynomial is 24. 3) a always equals 1. Find the sum of the c values of all polynomials in Bruce's set. This polynomial is true for (-3,27),(-7,31)... I don't know how to continue and find sum of c values of all polynomials.

First, look at the roots r₁ and r₂
\(\displaystyle r_1 = \frac{-b\, +\, \sqrt{b^2\, -\, 4\, c}}{2}\)
and
\(\displaystyle r_2 = \frac{-b\, -\, \sqrt{b^2\, -\, 4\, c}}{2}\)
since a is always equal to 1.

First, try to reduce the problem. That is, can we say anything definite about b. From (2) we know that the sum of the roots is 24 so what does that say about b.

Given b [from above] and the fact that all roots are prime and, as a corollary, that all roots are positive, what restrictions does that put on c?

Can you go from there?

 

[narinder]

Yes.. Thank you so much.