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Which of the following polynomial functions have exactly 5 roots, including all real and imaginary roots?Responsesf(x)=5x4+x3−x2+x−5𝑓(𝑥)=5𝑥4+𝑥3−𝑥2+𝑥−5f of x is equal to 5 x to the 4th power plus x cubed minus x squared plus x minus 5f(x)=4x5+3x4+2x3+x2+x𝑓(𝑥)=4𝑥5+3𝑥4+2𝑥3+𝑥2+𝑥f of x is equal to 4 x to the 5th power plus 3 x to the 4th power plus 2 x raised to the 3 power plus x squared plus xf(x)=5x6−5x4+5x2−5𝑓(𝑥)=5𝑥6−5𝑥4+5𝑥2−5f of x is equal to 5 x to the 6th power minus 5 x to the 4th power plus 5 x squared minus 5f(x)=4x4−3x3+2x2−1

Question

Which of the following polynomial functions have exactly 5 roots, including all real and imaginary roots?Responsesf(x)=5x4+x3−x2+x−5𝑓(𝑥)=5𝑥4+𝑥3−𝑥2+𝑥−5f of x is equal to 5 x to the 4th power plus x cubed minus x squared plus x minus 5f(x)=4x5+3x4+2x3+x2+x𝑓(𝑥)=4𝑥5+3𝑥4+2𝑥3+𝑥2+𝑥f of x is equal to 4 x to the 5th power plus 3 x to the 4th power plus 2 x raised to the 3 power plus x squared plus xf(x)=5x6−5x4+5x2−5𝑓(𝑥)=5𝑥6−5𝑥4+5𝑥2−5f of x is equal to 5 x to the 6th power minus 5 x to the 4th power plus 5 x squared minus 5f(x)=4x4−3x3+2x2−1

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Solution

Para determinar cuál de las funciones polinómicas tiene exactamente 5 raíces, incluyendo todas las raíces reales e imaginarias, debemos observar el grado de cada polinomio. El grado de un polinomio es el exponente más alto de la variable x x en la función, y este grado nos indica el número máximo de raíces (reales e imaginarias) que puede tener el polinomio.

Analicemos cada función:

  1. f(x)=5x4+x3x2+x5 f(x) = 5x^4 + x^3 - x^2 + x - 5

    • El grado de este polinomio es 4 (el exponente más alto es 4).
    • Un polinomio de grado 4 puede tener hasta 4 raíces.
    • No tiene exactamente 5 raíces.
  2. ( f(x) = 4x^5 + 3x^4 +

This problem has been solved

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