Question

zad.1

rozłoz wielomian na czynniki;

a)x3-4x-2x+8

b)x3-4x+3x2-12

c)-x3+5x2-x+5

d)w(x)=-4x4+26x3-12x2

zad.2

znajdz pierwiastki rownan;

a)x3+2x2=3x

b)x3+3x2-x-3=0

c)(x-1)(x2+6x+9)=0

d)(x+1)(x2-6x+9)=0

e)x5-5x3+4x=0

zad.3oblicz wartosc wielomianuw(x)dla x=-1

w(x)=-x4-2x3+x2-3x+1

Answer 1

zad.1

 

a)x³-4x²-2x+8=

=x²(x-4)-2(x-4)=

=(x-4)(x²-2)=

=(x-4)(x-√2)(x+√2)

 

b)x³-4x+3x²-12=

=x³+3x²-4x-12=

=x²(x+3)-4(x+3)=

=(x+3)(x²-4)=

=(x+3)(x-2)(x+2)

 

c)-x³+5x²-x+5=

=x²(-x+5)+1(-x+5)=

=(-x+5)(x²+1)

 

d)-4x⁴+26x³-12x²=

=2x²(-2x²+13x-6x)

Δ=13²-4*(-2)*(-6)=169-48=121

√Δ=11

x₁=(-13+11)/-4=½

x₂=(-13-11)/-4=6

 

=-4x²(x-½)(x-6)

 

zad.2

 

a)x³+2x²=3x

x³+2x²-3x=0

x(x²+2x-3)=0

Δ=2²-4*1*(-3)=4+12=16

√Δ=4

x₁=(-2+4)/2=1

x₂=(-2-4)/2=-3

 

x=0 ∨ x=1 ∨ x=-3

 

b)x³+3x²-x-3=0

x²(x+3)-1(x+3)=0

(x+3)(x²-1)=0

(x+3)(x+1)(x-1)=0

 

x=-3  x=-1 ∨ x=1

 

c)(x-1)(x²+6x+9)=0

Δ=6²-4*1*9=36-36=0

x=-6/2=-3

 

x=1 ∨ x=-3

 

d)(x+1)(x²-6x+9)=0

Δ=6²-4*1*9=36-36=0

x=-6/2=-3

 

x=-1 ∨ x=-3

 

 

zad.3

w(-1)=-(-1)⁴-2*(-1)³+(-1)²-3*(-1)+1=

=-1 +2 +1 +3 +1 =

= 6