Proving that not all roots of a cubic polynomial are real

Question

Consider a cubic polynomial $$p(x)=ax^3+bx^2+cx+d$$$a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even.The question is to prove that not all roots of $p(x)$ can be real.

I tried differentiating p(x) to get $$p'(x)=3ax^2+2bx+c$$Now for the cubic equation to have nonreal roots $4b^2-12ac<0$.But I donot see how I can apply the conditions as given in the problem.Any hints to proceed shall be highly appreciated.Thanks.

@Rohan But the OP is asking about real vs. complex roots, not rational vs. irrational. — Viktor Vaughn, Feb 16 '17 at 08:01
Does this help? http://math.stackexchange.com/questions/1393869/conditions-for-distinct-real-roots-of-cubic-polynomials — rookie, Feb 16 '17 at 08:02

Henry · Answer 1 · 2017-02-16T08:29:43.227

1

An example with $4b^2-12ac\gt 0$ and the odd/even conditions is $a=d=1, b=c=4$, i.e. $$x^3 +4 x^2 +4 x+1=0$$ which has the three real roots $-1,-\frac32 +\sqrt{{\frac52}},-\frac32 -\sqrt{{\frac52}}$ so this is a counterexample

As @Rohan suggests, the question should probably be about rational roots

edited Feb 16 '17 at 08:29

answered Feb 16 '17 at 08:14

Henry

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Proving that not all roots of a cubic polynomial are real

1 Answers1