Find all real and complex zeros of h(x)

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SUMMARY

The discussion focuses on finding all real and complex zeros of the cubic function h(x) = x^3 + 2x^2 - 16. Participants suggest using the Rational Root Theorem to identify potential roots, specifically the integers that divide -16, which include ±1, ±2, ±4, ±8, and ±16. The initial attempt at factoring was inadequate, and the recommendation is to express the polynomial in the form (x - r)(x^2 + lower degree terms) for further simplification. This approach is essential for solving cubic equations effectively.

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  • Understanding of cubic functions and their properties
  • Familiarity with the Rational Root Theorem
  • Basic polynomial factoring techniques
  • Knowledge of complex numbers and their representation
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  • Practice factoring cubic polynomials
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Students studying algebra, particularly those tackling polynomial equations, as well as educators looking for effective teaching strategies for cubic functions and their zeros.

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Homework Statement


It's asking me to find all the real and complex zeros of the function x^3 + 2x^2 - 16.


Homework Equations





The Attempt at a Solution


I have tried factoring the first 2 terms and i come up with x2(x+2) - 16 but I don't know where to go from there. Any help would be appreciated.
 
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FrugalIntelle said:

Homework Statement


It's asking me to find all the real and complex zeros of the function x^3 + 2x^2 - 16.


Homework Equations





The Attempt at a Solution


I have tried factoring the first 2 terms and i come up with x2(x+2) - 16 but I don't know where to go from there. Any help would be appreciated.

This is not really a very good start. Instead you should see if you can find a number r such that x3 + 2x2 - 16 = (x - r) * (x2 + lower degree terms).

The Rational Root Theorem (you can search for this on the web) says that if r is a root of your cubic polynomial, it has to be a number that evenly divides -16. The only candidates are ±1, ±2, ±4, ±8, and ±16. Does your textbook mention this theorem? Does your textbook show any examples of similar problems? Do you read your textbook?
 

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