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Ejercicio IV. Dados los polinomios:
[tex]\[
\begin{array}{l}
P(x)=x^4-2x^2-6x-1 \\
Q(x)=x^3-6x^2+4 \\
R(x)=2x^4-2x-2
\end{array}
\][/tex]
¿Cuál es la suma de [tex]\( P(x) + 2Q(x) - R(x) \)[/tex] ?

Sagot :

GinnyAnswer
Para resolver el ejercicio IV, necesitamos encontrar la suma de los polinomios [tex]\(P(x) + 2Q(x) - R(x)\)[/tex] dados:

1. [tex]\( P(x) = x^4 - 2x^2 - 6x - 1 \)[/tex]
2. [tex]\( Q(x) = x^3 - 6x^2 + 4 \)[/tex]
3. [tex]\( R(x) = 2x^4 - 2x - 2 \)[/tex]

En primer lugar, calculamos [tex]\(2Q(x)\)[/tex]:

[tex]\[ 2Q(x) = 2(x^3 - 6x^2 + 4) = 2x^3 - 12x^2 + 8 \][/tex]

Luego, sumamos [tex]\(P(x)\)[/tex], [tex]\(2Q(x)\)[/tex], y restamos [tex]\(R(x)\)[/tex]:

[tex]\[ P(x) + 2Q(x) - R(x) = (x^4 - 2x^2 - 6x - 1) + (2x^3 - 12x^2 + 8) - (2x^4 - 2x - 2) \][/tex]

Ahora realizamos la suma y resta término a término agrupando los coeficientes según las potencias de [tex]\(x\)[/tex]:

[tex]\[ \begin{aligned} P(x) + 2Q(x) - R(x) &= (x^4 + 2x^3 - 2x^2 - 6x - 1) \\ &\quad + (2x^3 - 12x^2 + 8) \\ &\quad - (2x^4 - 2x - 2) \\ \end{aligned} \][/tex]

Agrupando los coeficientes según las potencias:

- Coeficientes de [tex]\(x^4\)[/tex]: [tex]\(1 - 2 = -1\)[/tex]
- Coeficientes de [tex]\(x^3\)[/tex]: [tex]\(2 + 2 = 2\)[/tex]
- Coeficientes de [tex]\(x^2\)[/tex]: [tex]\(-2 - 12 = -14\)[/tex]
- Coeficientes de [tex]\(x\)[/tex]: [tex]\(-6 + 2 = -4\)[/tex]
- Coeficientes independientes: [tex]\(-1 + 8 + 2 = 9\)[/tex]

Por lo tanto, el polinomio resultante de la suma es:

[tex]\[ P(x) + 2Q(x) - R(x) = -x^4 + 2x^3 - 14x^2 - 4x + 9 \][/tex]

En forma de coeficientes, la respuesta es:

[tex]\[ \boxed{[-1, 2, -14, -4, 9]} \][/tex]

Este es el polinomio resultado de [tex]\( P(x) + 2Q(x) - R(x) \)[/tex].
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