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The ubiquity of polynomials and their ability to characterize complex patterns let us better understand generalizations, theorems, and elegant paths to solutions that they provide. We strive to showcase the true beauty of polynomials through a well-thought collection of problems from mathematics competitions and intuitive lectures that follow the sub-topics. Thus, we present a view of polynomials that incorporates various techniques paired with the favorite themes that show up in math contests.
Preface
Basic Properties of Polynomials - Part I
Identities
The coefficients of xd in polynomial products
Factoring and its implications
Values of polynomials
Division, GCD of polynomials
The composition of polynomials
Odd and even polynomials
Basic Properties of Polynomials - Part II
Polynomial roots
Integer and rational roots of polynomials
Intermediate value theorem, increasing and decreasing polynomials
Second degree polynomials
The form ax2+bx+c
The Discriminant
Roots
Vieta's formulas
Solving Inequalities
Miscellaneous problems
More advanced problems
Third degree polynomials
Roots and graph
Vieta's formulas
More advanced problems
Fourth degree polynomials
Solving equations
Vieta's formulas
Number of real roots and graph
Miscellaneous
On roots of polynomials - elementary problems
Vieta's formulas in the general case
Inequalities between coefficients and roots
Miscellaneous problems
Number theory and polynomials
Number theory and low degree polynomials
P(a)-P(b)
Introductory problems
Advanced problems
Solutions to introductory problems
Solutions to advanced problems
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