Question

Wielomiany

1. Rozłóż na czynniki wielomian W (x) = 2x⁴ - x³ - 3x²

2. Rozłóż na czynniki wielomian W (x) = 3x³ + 24

3. Rozwiąż równanie x³ - 3x² - 4x + 12 = 0

Answer 1

1.

W (x) = 2x⁴  - x³ - 3x² =

=x²(2x²-x-3)=

=2x²(x-1½)(x+1)

 

2x²-x-3=0

Δ=(-1)²-4*2*(-3)1+24=25

√Δ=5

x₁=(1+5)/4=1½

x₂=(1-5)/4=-1

 

2.W (x) = 3x³ + 24=

=3(x³+8)=

=3(x+2)(x²-2x+4)

 

x²-2x+4=0

Δ=(-2)²-4*1*4= -12<0

 

3.x³ - 3x² - 4x + 12 = 0

x²(x-3)-4 (x -3)=0

(x²-4)(x-3)=0

x²=4

x=2 ∨x=-2

 

x-3=0

x=3

Answer 2

z.1

W(x) = 2x⁴ - x³ - 3 x² = x² (2 x² - x - 3)

Δ = 1 - 4*2*(-3) = 1 +24 = 25

√Δ  = 5

x1 = [ 1 - 5]/4 = -1

x2 = [ 1 + 5]/4 = 1,5

czyli 2x² -x -3 = 2( x +1)(x - 1,5)

zatem W(x) = 2 x² (x +1)( x -1,5)

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z.2

W(x) = 3 x³ + 24 = 3( x³ + 2³) = 3(x +2)(x² -2x +4)

Δ = 4 - 4*1*4 < 0

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z.3

x³ + 3 x² - 4x + 12 = 0

x²(x + 3) - 4(x +3) = 0

(x² - 4)( x +3) = 0

(x +2)(x -2)(x +3) = 0

x1 = -3

x2 = -2

x3 = 2

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x³ - 3 x² - 4x + 12 = 0

x² (x -3) - 4(x -3) = 0

(x² - 4)( x -3) = 0

(x +2)(x -2)(x -3) = 0

x+2 = 0  lub  x -2  = 0  lub x -3 = 0

x = -2  lub x = 2  lub  x = 3

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