Question

Zad.6

Rozłóż wielomiany na czynniki.

a) w(x)=x^{4}-16

b) w(x)=

c) w(x)=

d) w(x)=

e) w(x)=

f) w(x)=

g) w(x)=

Answer 1

a) w(x)=x^{4}-16

(x²-4)(x²+4)

b) w(x)=

x⁴(2x²+3x-2)=x⁴(x-½)(x+2)

c) w(x)=

4x³(x²+2x+1)=4x³(x+1)²

d) w(x)=

(x-2)(x²+2x+4)

e) w(x)=

x²(x-5)+2(x-5)=(x-5)(x²+2)

f) w(x)=

x³(x-1)-8x(1x-1)=(x-1)(x³-8x)(x-1)x(x-8)

g) w(x)=

x(9x²-1)=x(3x-1)(3x+1)

 

Answer 2

a) w(x)= x⁴-16= (x-4)(x+4)

 

b) w(x)= 2x⁶+ 3x⁵- 2x⁴= x⁴(2x²+3x+2)

Δ=9+16=25

√Δ=5

x₁=(-3+5)/4=½

x₂=(-3-5)/4= -2

w(x)= 2x⁶+ 3x⁵- 2x⁴= x⁴(x-½)(x+2)

 

c) w(x)=4x⁵+8x⁴+4x³= 4x³(x²+2x+1)= 4x³(x+1)²

 

d) w(x)=x³-8= (x-2)(x²+2x+4)

 

e) w(x)= x³-5x²+2x-10

{-10, -5, -2, -1, 1, 2, 5, 10

 

w(-10)= -1000-500-20-10≠0

w(-5)= -125-125-10-10≠0

w(-2)= -8-20-4-10≠0

w(-1)= -1-5-2-10≠0

w(1)= 1-1+2-10≠0

w(2)= 8-20+4-10≠0

w(5)= 125-125+10-10=0

x²+2

(x³-5x²+2x-10):(x-5)

-x³+5x²

=   =   2x-10

-2x+10

=   =

w(x)= x³-5x²+2x-10= (x-5)(x²+2)

 

f) w(x)=x⁴-x³-8x²+8x= x(x³-x²-8x+8)

{-8, -4, -2, -1, 1, 2, 4, 8

 

w(-8)= x³-x²-8x+8= -512-64-64+8≠0

w(-4)= x³-x²-8x+8= -64-8-32+8≠0

w(-2)= x³-x²-8x+8= -8-4-16+8≠0

w(-1)= x³-x²-8x+8= -1-1-8+8≠0

w(1)= x³-x²-8x+8= 1-1-8+8=0

 

x²-8

(x³-x²-8x+8):(x-1)

-x³+x²

=   =  -8x+8

8x-8

=  =

x²-8=(x-2√2)(x+2√2)

 

w(x)=x⁴-x³-8x²+8x= x(x-1)(x-2√2)(x+2√2)

 

g) w(x)=9x³-x= x(9x²-1)= x(3x-1)(3x+1)