Question

Rożłóż wielomian w na czynniki,grupujac jego wyrazy (przyklady w załaczniku)

Answer 1

a]x⁴+2x³-8x-16=x³(x+2)-8(x+2)=(x³-8)(x+2)=(x-2)((x²+2x+4)(x+2)

b]14x³-7x²+4x-2=7x²(2x-1)+2(2x-1)=(7x²+2)(2x-1)

c]2x³-6x²+5x-15=2x²(x-3)+5(x-3)=(2x²+5)(x-3)

d]x⁴-3x³+x-3=x³(x-3)+(x-3)=(x³+1)(x-3)=(x+1)(x²-x+1)(x-3)

e]½x³-⅙x²-3x+1=⅙x²(3x-1)-(3x-1)=(⅙x²-1)(3x-1)=(√6/6 x+1)(√6/6x-1)(3x-1)

f]⅔x³-3x²-6x+27=x²(⅔x-3)-9(⅔x-3)=(x²-9)(⅔x-3)=(x+3)(x-3)(⅔x-3)

g]x³-√2x²+√2x-2=x²(x-√2)+√2(x-√2)=(x²+√2)(x-√2)

h]x⁵+x⁴-2x³-2x²+x+1=x⁴(x+1)-2x²(x+1)+(x+1)=(x⁴-2x²+1)(x+1)=(x²-1)²(x+1)=(x²-1)(x²-1)(x+1)=(x+1)(x-1)(x+1)(x-1)(x+1)=(x+1)³(x-1)² bo:

 

x²=t

t²-2t+1=0

Δ=b²-4ac=4-4=0

t₀=-b/2a=2/2=1

x²=1

x=1 lub x=-1