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5. Restar los siguientes polinomios:
[tex]\[ P(x) = x^4 - x^3 - x^2 + 2x + 2 \][/tex]
[tex]\[ Q(x) = 2x^2 + 3x^3 + 4x^4 - 5x + 5 \][/tex]

6. Calcular el valor numérico de \( P(x) \) para los siguientes valores:
a) \( x = 1 \)
b) \( x = -1 \)
c) \( x = \frac{2}{3} \)
d) \( x = -3 \)

[tex]\[ P(x) = \frac{x}{2} - 3x + 4x^2 - 5x^3 - \frac{2x^4}{3} + \frac{5}{4} \][/tex]

7. Dados los polinomios:
[tex]\[ P(x) = 4x^2 - x + 2 \][/tex]
[tex]\[ Q(x) = x^3 + x - 1 \][/tex]
[tex]\[ R(x) = 2x - 1 \][/tex]

Hallar:
a) \( P(x) + Q(x) \)
b) \( P(x) + R(x) \)
c) \( Q(x) \cdot R(x) \)
d) \( P(x) \cdot Q(x) \)
e) \( P(x) : R(x) \)
f) \( Q(x) : R(x) \)
g) El resto de la división de \( P(x) \) por \( x - 1 \)
h) \( P(-1) \)
i) [tex]\( P(-2) + [Q(-2)]^2 \)[/tex]

Sagot :

GinnyAnswer
¡Claro! Vamos a resolver cada parte de la pregunta paso a paso.

5) Restar los siguientes polinomios:
Dados los polinomios

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

y

[tex]\[ Q(x) = 2x^2 + 3x^3 + 4x^4 - 5x + 5 \][/tex]

Resta los dos polinomios:

[tex]\[ R(x) = P(x) - Q(x) \][/tex]

[tex]\[ R(x) = (x^4 - x^3 - x^2 + 2x + 2) - (2x^2 + 3x^3 + 4x^4 - 5x + 5) \][/tex]

[tex]\[ R(x) = x^4 - x^3 - x^2 + 2x + 2 - 2x^2 - 3x^3 - 4x^4 + 5x - 5 \][/tex]

Reorganizando términos:

[tex]\[ R(x) = x^4 - 4x^4 - x^3 - 3x^3 - x^2 - 2x^2 + 2x + 5x + 2 - 5 \][/tex]

[tex]\[ R(x) = -3x^4 - 4x^3 - 3x^2 + 7x - 3 \][/tex]

6) Calcular el valor numérico de \(P(x)\) para los siguientes valores:

Dado el polinomio

[tex]\[ P(x) = \frac{x}{2} - 3x + 4x^2 - 5x^3 - \frac{2x^4}{3} + \frac{5}{4} \][/tex]

Evaluamos para cada valor de \(x\):

a) \( x = 1 \)

[tex]\[ P(1) = \frac{1}{2} - 3(1) + 4(1^2) - 5(1^3) - \frac{2(1^4)}{3} + \frac{5}{4} = -2.91666666666667 \][/tex]

b) \( x = -1 \)

[tex]\[ P(-1) = \frac{-1}{2} - 3(-1) + 4(-1)^2 - 5(-1)^3 - \frac{2(-1^4)}{3} + \frac{5}{4} = 12.0833333333333 \][/tex]

c) \( x = \frac{2}{3} \)

[tex]\[ P\left(\frac{2}{3}\right) = \frac{\frac{2}{3}}{2} - 3\left(\frac{2}{3}\right) + 4\left(\frac{2}{3}\right)^2 - 5\left(\frac{2}{3}\right)^3 - \frac{2\left(\frac{2}{3}\right)^4}{3} + \frac{5}{4} = -0.252057613168724 \][/tex]

d) \( x = -3 \)

[tex]\[ P(-3) = \frac{-3}{2} - 3(-3) + 4(-3)^2 - 5(-3)^3 - \frac{2(-3)^4}{3} + \frac{5}{4} = 125.750000000000 \][/tex]

7) Dados los polinomios:
[tex]\[ P(x) = 4x^2 - x + 2 \][/tex]
[tex]\[ Q(x) = x^3 + x - 1 \][/tex]
[tex]\[ R(x) = 2x - 1 \][/tex]

Hallar:

a) \(P(x) + Q(x)\)

[tex]\[ (4x^2 - x + 2) + (x^3 + x - 1) = x^3 + 4x^2 + 1\][/tex]

b) \(P(x) + R(x)\)

[tex]\[ (4x^2 - x + 2) + (2x - 1) = 4x^2 + x + 1 \][/tex]

c) \(Q(x) \cdot R(x)\)

[tex]\[ (x^3 + x - 1) \cdot (2x - 1) = (2x - 1)(x^3 + x - 1) \][/tex]

d) \(P(x) \cdot Q(x)\)

[tex]\[ (4x^2 - x + 2) \cdot (x^3 + x - 1) = (4x^2 - x + 2)(x^3 + x - 1) \][/tex]

e) \(P(x) / R(x)\)

[tex]\[ \frac{4x^2 - x + 2}{2x - 1} = 2x + \frac{1}{2} + \text{residuo } \frac{5}{2} \][/tex]

f) \(Q(x) / R(x)\)

[tex]\[ \frac{x^3 + x - 1}{2x - 1} = \frac{x^2}{2} + \frac{x}{4} + \frac{5}{8} + \text{residuo } \frac{-3}{8} \][/tex]

g) El resto de la división de \(P(x)\) por \(x - 1\)

[tex]\[ \frac{4x^2 - x + 2}{x - 1} \rightarrow \text{residuo} = 5 \][/tex]

h) \(P(-1)\)

[tex]\[ P(-1) = 4(-1)^2 - (-1) + 2 = 7 \][/tex]

i) \(P(-2) + [Q(-2)]^2\)

[tex]\[ P(-2) = 4(-2)^2 - (-2) + 2 = 22 \][/tex]
[tex]\[ Q(-2) = (-2)^3 + (-2) - 1 = -11 \][/tex]
[tex]\[ P(-2) + [Q(-2)]^2 = 22 + (-11)^2 = 141 \][/tex]
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