1. Rozłóż wielomian w na czynniki
a. w(x)= -x⁶+3x⁵+10x⁴
b. w(x)=x³+√2x²+x
c. w(x)=-9x⁴+25x²
d. w(x)=2x⁵-18x³
a. w(x)= -x⁶+3x⁵+10x⁴ = - x^4(x^2 - 3x - 10) = -x^4(x + 2)(x - 5)
Δ = 9 - 4 * 1 * (-10) = 9 + 40 = 49
√Δ = 7
x1 = (3 - 7)/2 = -4/2 = -2
x2 = (3 + 7)/2 = 10/2 = 5
b. w(x)=x³+√2x²+x = x(x² + √2x + 1)
Δ = 2 - 4 * 1 * 1 = 2 - 4 = -2 < 0
c. w(x)=-9x⁴+25x² = -x²(9x² - 25) = -x² (3x - 5)(3x + 5)
d. w(x)=2x⁵-18x³ = 2x³(x² - 9) = 2x³(x - 3)(x +3)
a/ W(x)= -x⁶+3x⁵+10x⁴= x⁴(-x²+3x+10)=... dla -x²+3x+10 Δ=9-4·(-1)·10=9+40=49, √Δ=7
x₁=(-3-7)/-2=-10/-2=5
x₂=(-3+7)/-2=4/-2=-2
...=x⁴(x-5)(x+2)
b/ W(x)=x³+√2x²+x=x(x²+√2x+1)=... dla x²+√2x+1 Δ=2-4·1·1=2-4=-2<0
wielomian nie da sie dalej rozlozyc na czynniki
c/ W(x)=-9x⁴+25 x²=-9x²(x²-25/9)= -9x²(x+5/3)(x-5/3) , bo a²-b²=(a+b)(a-b)
d/ W(x)= 2x⁵-18x³= 2x³(x²-9)=2x³(x+3)(x-3)
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a. w(x)= -x⁶+3x⁵+10x⁴ = - x^4(x^2 - 3x - 10) = -x^4(x + 2)(x - 5)
Δ = 9 - 4 * 1 * (-10) = 9 + 40 = 49
√Δ = 7
x1 = (3 - 7)/2 = -4/2 = -2
x2 = (3 + 7)/2 = 10/2 = 5
b. w(x)=x³+√2x²+x = x(x² + √2x + 1)
Δ = 2 - 4 * 1 * 1 = 2 - 4 = -2 < 0
c. w(x)=-9x⁴+25x² = -x²(9x² - 25) = -x² (3x - 5)(3x + 5)
d. w(x)=2x⁵-18x³ = 2x³(x² - 9) = 2x³(x - 3)(x +3)
a/ W(x)= -x⁶+3x⁵+10x⁴= x⁴(-x²+3x+10)=... dla -x²+3x+10 Δ=9-4·(-1)·10=9+40=49, √Δ=7
x₁=(-3-7)/-2=-10/-2=5
x₂=(-3+7)/-2=4/-2=-2
...=x⁴(x-5)(x+2)
b/ W(x)=x³+√2x²+x=x(x²+√2x+1)=... dla x²+√2x+1 Δ=2-4·1·1=2-4=-2<0
wielomian nie da sie dalej rozlozyc na czynniki
c/ W(x)=-9x⁴+25 x²=-9x²(x²-25/9)= -9x²(x+5/3)(x-5/3) , bo a²-b²=(a+b)(a-b)
d/ W(x)= 2x⁵-18x³= 2x³(x²-9)=2x³(x+3)(x-3)