I am preparing for a mathematics aptitude test and one of the topics that is heavily tested is Polynomials. Last year the following problems were asked:

1.Consider the polynomial $ax^3+bx^2+cx+d$ where $a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even.Prove that not all of its roots are rational.

2.If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.

I request the community members if they are aware of books that can prepare one to tackle problems of such difficulty. Also please write the background knowledge one must have before reading the referred books.

  • $\begingroup$ You should study the theorem of Vieta and the discriminant of a polynomial (perhaps helpful for the second question) $\endgroup$ – Peter Nov 9 '16 at 18:22
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    $\begingroup$ I suggest you go through the book 'Challenges and thrills of pre-college mathematics'. Not only is it recommended for the ISI entrance, it has a large collection of unsolved problems. Sometimes ISI questions are directly set from these exercises. But honestly, the theory part isn't that illuminating. I would rather suggest an undergrad textbook called 'Classical algebra' by SK Mapa for the exact theory. Besides this, you could always get hold of the relatively difficult 'Problem solving strategies' by Arthur Engels. $\endgroup$ – StubbornAtom Nov 9 '16 at 18:35
  • $\begingroup$ @StubbornAtom I have heard about E J Barbeau (not sure of the spelling). Is it good enough? Also, could you please give a booklist for ISI entrance. $\endgroup$ – model_checker Nov 9 '16 at 19:26
  • $\begingroup$ That's another Springer book that you can download. I haven't used it and I don't think it would be necessary after having gone through the previous books. As for the recommended books for ISI, they had that in their website. But you don't really need every single book to prepare. $\endgroup$ – StubbornAtom Nov 10 '16 at 3:10

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