Necessary and sufficient conditions for $ax^2+bx+c$ to be factorizable...

Question

What are necessary and sufficient conditions for $ax^2+bx+c$ to be factorizable involving just integer coefficients?

We already know that this expression is factorizable if $b^2-4ac$ is a perfect square,($ax^2+bx+c$ is factorizable if...)but in this case can we be sure that all the coefficients are just integers??

Assuming you mean factorizable over $\mathbb{Q}$, a necessary condition is given by Eisenstein's criterion. — MathematicsStudent1122, Jul 06 '16 at 20:46
@MathematicsStudent1122 No,I mean just factorizable over $\Bbb{Z}$ — Hamid Reza Ebrahimi, Jul 06 '16 at 20:48
By Gauss's lemma, if you can factor a nonconstant polynomial (with integer coefficients) over the rationals, you can factor it over the integers. — Robert Israel, Jul 06 '16 at 20:49
@RobertIsrael Is there any algorithm for converting a rational factorization of a polynomial into an integer factorization? — Hamid Reza Ebrahimi, Jul 06 '16 at 21:20
Put everything over a common denominator, and take out common integer factors of the numerators. Denominators should cancel with the factors of the numerators. — Robert Israel, Jul 06 '16 at 21:40
@RobertIsrael What about this: $(x+\frac13)(x+1)=(\frac{3x+1}{3})(x+1)$?!! — Hamid Reza Ebrahimi, Jul 07 '16 at 04:59
$x^2 + \frac{4}{3} x + \frac{1}{3}$ does not have integer coefficients. — Robert Israel, Jul 07 '16 at 15:27

score 1 · Answer 1 · answered Jul 06 '16 at 21:05

1

Robert Israel's comment:

By Gauss's lemma, if you can factor a nonconstant polynomial (with integer coefficients) over the rationals, you can factor it over the integers.

And as you point out, it factors over the rationals just in case $b^2 - 4ac$ is a perfect square. Therefore it factors over the integers just in case $b^2 - 4ac$ is a perfect square.

answered Jul 06 '16 at 21:05

Caleb Stanford

45,895

Yes,in fact my question had been already answered by Gauss's lemma! which I didn't know about. – Hamid Reza Ebrahimi Jul 06 '16 at 21:09

Necessary and sufficient conditions for $ax^2+bx+c$ to be factorizable...

1 Answers1