Question

x³ + 3x² + x-2 ÷ x² + x - 1​

Answer 1

Respuesta:

P(x) = 5x2 - 7x + 3

Q(x) = -5x2 + 2x

R(x) = x3 + x2 + 2

a) P(x) + Q(x)

a) P(x) + Q(x) = 5x2 - 7x + 3 - 5x2 + 2x =

= 5x2 - 5x2 - 7x + 2x + 3 = -5x + 3

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

b) P(x) + R(x) = 5x2 - 7x + 3 + x3 + x2 + 2 =

= x3 + 5x2 + x2 - 7x + 3 + 2 = x3 + 6x2 - 7x + 5

c) Q(x) + R(x)

c) Q(x) + R(x) = -5x2 + 2x + x3 + x2 + 2 =

= x3 - 5x2 + x2 + 2x + 2 = x3 - 4x2 + 2x + 2

2)   Hallar el polinomio diferencia entre:

a) P(x) = x4 + x2 + 2   y   Q(x) = x3 - 2x2 - 5x + 6

a) P(x) - Q(x) = x4 + x2 + 2 - (x3 - 2x2 - 5x + 6) =

= x4 + x2 + 2 - x3 + 2x2 + 5x - 6 = x4 - x3 + 3x2 + 5x - 4

b) P(x) = x3 + x2 - x + 1   y   Q(x) = 2x2 + 3x + 4

b) P(x) - Q(x) = x3 + x2 - x + 1 - (2x2 + 3x + 4) =

= x3 + x2 - x + 1 - 2x2 - 3x - 4 = x3 - x2 - 4x - 3

c) P(x) = x5 - 2x3 + 4x2 - 6   y   Q(X) = x5 + x4 + 3x2 + 4x + 5

c) P(x) - Q(x) = x5 - 2x3 + 4x2 - 6 - (x5 + x4 + 3x2 + 4x + 5) =

= x5 - 2x3 + 4x2 - 6 - x5 - x4 - 3x2 - 4x - 5 = -x4 - 5x3 + x2 - 4x - 11

3)   Calcula y simplifica:

(x2 - 5x + 1) - (3x - 1) + (2x2 + 3x - 1) - (x3 + 2x - 5)

(x2 - 5x + 1) - (3x - 1) + (2x2 + 3x - 1) - (x3 + 2x - 5) =

= x2 - 5x + 1 - 3x + 1 + 2x2 + 3x - 1 - x3 - 2x + 5 = -x3 + 3x2 - 7x + 6

4)   Sean   P(x) = x2 - 4x + 2   y   Q(x) = 2x3 + x2 + 5.   Calcular:

a) -2P(x)

a) -2P(x) = -2(x2 - 4x + 2) = - 2x2 + 8x - 4

b) 4Q(x)

b) 4Q(x) = 4(2x3 + x2 + 5) = 8x3 + 4x2 + 20

c) 3P(x) - 2Q(x)

c) 3P(x) - 2Q(x) = 3(x2 - 4x + 2) - 2(2x3 + x2 + 5) =

= 3x2 - 12x + 6 - 4x3 - 2x2 - 10 = -4x3 + x2 - 12x - 4

También podemos resolverlo de esta otra forma:

   3P(x)         =              3x2 - 12x + 6

 -2Q(x)         =   - 4x3 - 2x2          - 10 ________________________________

3P(x) - 2Q(x)  =  - 4x3  + x2  - 12x  - 4

Multiplicación de polinomios:

1)   Hallar los siguientes productos:

a) (-2x2)(x5 - 4x2 + 3x + 1)

a) (-2x2)(x5 - 4x2 + 3x + 1) = - 2x7 + 8x4 - 6x3 - 2x2

b) (x2 - 1)(5x5)

b) (x2 - 1)(5x5) = 5x7 - 5x5

c) (-3x3)(2x4 - 3x3 + 2x - x + 3)

c) (-3x3)(2x4 - 3x3 + 2x - x + 3) = - 6x7 + 9x6 - 6x4 + 3x4 - 9x3

2)   Hallar los siguientes productos:

a) P(x) = 5x2 + 3x - 1   y   Q(x) = x + 2

a) P(x)·Q(x) = (5x2 + 3x - 1)·(x + 2) =

= (5x2 + 3x - 1)x + (5x2 + 3x - 1)2 =

= 5x3 + 3x2 - x + 102 + 6x - 2 = 5x3 + 13x2 + 5x - 2

b) P(x) = x3 + 1   y   Q(x) = x2 + x + 1

b) P(x)·Q(x) = (x3 + 1)·(x2 + x + 1) =

= x3(x2 + x + 1) + (x2 + x + 1) = x5 + x4 + x3 + x2 + x + 1

c) P(x) = x3 + 2x2 + x + 1   y   Q(x) = x2 - 1

c) P(x)·Q(x) = (x3 + 2x2 + x + 1)·(x2 - 1) =

= (x3 + 2x2 + x + 1)x2 - (x3 + 2x2 + x + 1) =

= x5 + 2x4 + x3 + x2 - x3 - 2x2 - x - 1 = x5 + 2x4 - x2 - x - 1

d) P(x) = - x3 + 4x - 3   y   Q(x) = x2 + 3x + 4

d) P(x)·Q(x) = (- x3 + 4x - 3)·(x2 + 3x + 4) =

= - x3(x2 + 3x + 4) + 4x(x2 + 3x + 4) - 3(x2 + 3x + 4) =

= - x5 - 3x4 - 4x3 + 4x3 + 12x2 + 16x - 3x2 - 9x - 12 = - x5 - 3x4 + 9x2 + 7x - 12

Explicación paso a paso:

espero que te ayude