Challenge for Polynomial with Complex Coefficients

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SUMMARY

The discussion centers on proving that the roots of the quadratic polynomial $ax^2 + bx + c$, where $a$ and $b$ are non-zero complex coefficients, lie within the region defined by the inequality $|x| \le \left|\dfrac{b}{a}\right| + \left|\dfrac{c}{b}\right|$. The proof leverages properties of complex numbers and the triangle inequality to establish the bounds on the roots effectively. This result is crucial for understanding the behavior of polynomials with complex coefficients.

PREREQUISITES
  • Complex number theory
  • Quadratic equations and their properties
  • Triangle inequality in complex analysis
  • Understanding of polynomial roots
NEXT STEPS
  • Study the properties of complex coefficients in polynomials
  • Learn about the triangle inequality in the context of complex analysis
  • Explore the implications of root bounds in polynomial equations
  • Investigate the behavior of quadratic polynomials with varying coefficients
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Mathematicians, students studying complex analysis, and anyone interested in the properties of polynomial equations with complex coefficients.

anemone
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Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region

$|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.
 
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Hint:

Consider $|\sqrt{b^2-4ac}|$.
 

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