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The polynomial $p^3+ap^2+bp+13$ has three integer roots. Find the values of a and b.

I can't come up with anything at all, could someone show me a baby hint please?

Does it mean anything for a cubic to have three integer roots?


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Let the roots be $x,y,z$. Then $p^3 + ap^2 + bp + 13 = (p-x)(p-y)(p-z)$. –  Daniel Fischer Sep 8 '13 at 17:37

2 Answers 2

Just compute $(p-r)(p-s)(p-t)$ and compare coeficients with $p^3+ap^2+bp+13$. This gives Diophantine equations, i.e., $-(r+s+t)=a$, $rs + rt + st=b$, and $rst=-13$. A possible solution is $p^3 + 11p^2 - 25p + 13$, or $p^3 - 13p^2 - p + 13$, etc.

Comment: Yes, it means a lot for a cubic to have three integer roots. It even means a lot to have only real roots.

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Hint: consider the product of the roots. Three roots evidently means three different roots. Given what you know about the product, what could those roots be?

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