Need proof that a cubic equation has at least one real root

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    Cubic Proof Root
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SUMMARY

A cubic equation, defined as a polynomial of degree three, is guaranteed to have at least one real root due to the properties of polynomial functions. The discussion highlights the use of calculus, specifically continuity on the real number line and the application of Rolle's Theorem, to establish this fact. Additionally, it is noted that complex roots of real polynomials occur in conjugate pairs, which ensures that an odd-degree polynomial, such as a cubic, must have at least one unpaired real root.

PREREQUISITES
  • Understanding of cubic equations and polynomial functions
  • Knowledge of calculus, particularly continuity and Rolle's Theorem
  • Familiarity with complex numbers and their conjugates
  • Basic concepts of polynomial root behavior
NEXT STEPS
  • Study the proof of Rolle's Theorem and its applications in calculus
  • Explore the Fundamental Theorem of Algebra and its implications for polynomial roots
  • Learn about the behavior of polynomial functions at asymptotic limits
  • Investigate the properties of complex conjugates in relation to polynomial equations
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Students of mathematics, particularly those studying algebra and calculus, as well as educators seeking to explain the properties of polynomial equations and their roots.

lmamaths
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Hi,

A cubic equation has at least one real root.
Can someone help me to prove this?

Thx!
LMA
 
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U need calculus.It's not difficult.Use the continuitiy on \mathbb{R} and evaluate the 2 asymptotic limits.Then use the Rolle (IIRC) theorem...

Daniel.
 
I don't think you necessarily need that.

If w is a complex root of f(x) a real polynomial (ie one with real coeffs), then so is w* the conjugate of w, this means that the real poly (x-w)(x-w*) divides f(x) over the reals. Hence complex roots occur in conjugate pairs. A cubic has 3 (possibly complex) roots, so pairing up the complex ones (or any poly of odd degree) means there must be an odd number of unpaired real roots left.
 

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