Question

Rozłóż wielomian na czynniki:

a) w(x) = x do potegi 2 + 4x+4

b) w(x)= 9x do potegi2 - 6x+1

c) w(x) = x do potegi 4 - 2x do potegi 2 +1

d) w(x)= x do potegi 6 + 16x do potegi 3 +64

e) w(x) = x do potegi 4 - 16

f) w(x) = 1-25x do potegi 4

g) w(x) = x do potegi 4 - 5

h) w(x)= 1-9x do potegi 6

i) w(x) = x do potegi 3 - 1

j) w(x) = x do potegi 3+8

k) w(x) = 1-27x do potegi 3

l) w(x) = 1+ x do potegi 3 w liczniku i w mianowniku 8

Answer 1

a)   W(x) = x^2 + 4 x + 4 = ( x + 2)^2 = ( x + 2)*(x + 2)

b)   W(x) = 9 x^2 - 6 x + 1 = ( 3 x - 1)^2 = ( 3 x -1)*(3 x - 1)

c)  W(x) = x^4 -2 x^2 + 1 = ( x^2 - 1)^2 = [ ( x - 1)*( x + 1)]^2 =

             = ( x - 1)*(x -1)*(x + 1)*(x + 1)

d)  W(x) = x^6 + 16 x^3 + 64 = ( x^3 + 8)^2 = [ ( x + 2)*(x^2 - 2 x + 4)]^2 =

             = ( x + 2)*(x + 2)*( x ^2 - 2 x + 4)*( x^2 -2 x + 4)

                 bo  a^3 + b^3 = ( a + b)*( a^2 - a*b + b^2)

                 a = x ,  b = 2

e)  W(x) = x^4 - 16 = (x^2)^2 - 4^2 = ( x^2 - 4)*(x^2 + 4) = ( x -2)*(x + 2)*( x^2 + 4)

f)  W(x) = 1 - 25 x^4  = ( 1 - 5 x^2)*(1 + 5 x^2) = ( 1 - p(5) x)*(1 = p(5) x)*(1 + 5 x^2 )

          p(5)  - pierwiastek kwadratowy z 5

g)   W(x) = x^4 - 5 = ( x^2 - p(5))*( x^2 + p(5)) =

             = ( x - 5^(1/4))*( x + 5 ^(1/4))*( x^2 + p(5))

h)   W(x) = 1 - 9 x^6 = ( 1 - 3 x^3)*( 1 + 3 x^3) =

i )   W(x) = x^3 - 1 = ( x - 1)*( x^2 + x + 1)

j)    W(x) = x^3 + 8 = x^3 + 2^3 = ( x + 2)*(x^2 - 2x + 4)

k)   W(x) = 1 - 27 x^3 = 1 - (3x)^3 = ( 1 - 3x)*(1 + 3x + 9 x^2)

l)    W(x) = [ 1 + x^3]/ 8 = 1/8 + x^3/8 = ( 1/2)^3 + ( x/2)^3 =

             = ( 1/2 + x/2)*( 1/4 - x/4 + x^2/4 )

=========================================