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a)3x²/2x
x≠0
3x²/2x=3/2x
b)12y³/15y
y≠0
12y³/15y=4/5y²
c)15ab²/3a²b
a≠0, b≠0
15ab²/3a²b=5b/a
d)-4abc²/14a²bc
a≠0, b≠0, c≠0
-4abc²/14a²bc=-2c/7a
e)15a³b²c/3a²b³c²
a≠0, b≠0, c≠0
15a³b²c/3a²b³c²=5a/bc
f)3a-6b/a²-2ab
a²-2ab≠0
a(a-2b)≠0
a≠0, a≠2b
3a-6b/a²-2ab=3(a-2b)/a(a-2b)=3/a
g)5z²-3z/4z³-z²
4z³-z²≠0
z²(4z-1)≠0
z≠0, z≠1/4
5z²-3z/4z³-z²=z(5z-3)/z(4z²-z)=(5z-3)/(4z²-z)
h)4t⁴+10t³/6t²-8t³
6t²-8t³≠0
t²(6-8t)≠0
t≠0, t≠3/4
4t⁴+10t³/6t²-8t³=2t²(2t²+5t)/2t²(3-4t)=(2t²+5t)/(3-4t)
i)2a²-3ab/2ab-b²
2ab-b²≠0
b(2a-b)≠0
b≠0, 2a≠b
tego wyrażenia już się nie skróci bardziej
Ad.2
a)2x³+8x²+8x
x∈R
b)x²-x-2/x²+x-6
x²+x-6≠0
delta=1-4*1*6=25
√delta=5
x1=(-1-5)/2=-3
x2=(-1+5)/2=2
zatem x≠2 i x≈-3
x²-x-2/x²+x-6 = x²-x-2/(x-2)(x+3)
deltaLicznika=1-4*1*2=9
√deltaLicznika=3
x1=(1-3)/2=-1
x2=1+3/2=2
x²-x-2/x²+x-6 = x²-x-2/(x-2)(x+3)=(x+1)(x-2)/(x-2)(x+3)=x+1/x+3
c)x²-2x-8/2x²+3x-2
2x²+3x-2≠0
delta=9-4*2*(-2)=25
√delta=5
x1=(-3-5)/4=-2
x2=(-3+5)/4=1/2
x²-2x-8/2x²+3x-2=x²-2x-8/2(x+2)(x-1/2)=x²-2x-8/(x+2)(2x-1)
deltaLicznika=4-4*1*(-8)=4+32=36
√deltaLicznika=6
x1=(2-6)/2=-2
x2=(2+6)/2=4
zatem x≠-2 i x≈4
x²-2x-8/2x²+3x-2=x²-2x-8/2(x+2)(x-1/2)=x²-2x-8/(x+2)(2x-1)=(x-4)(x+2)/(x+2)(2x-1)=(x-4)/(2x-1)
d)3x²+8x-3/6x²-11x+3
6x²-11x+3≠0
delta=121-4*3*6=49
√delta=7
x1=(11-7)/12=1/3
x2=(11+7)/12=3/2
zatem x≠1/3 i x≈3/2
3x²+8x-3/6x²-11x+3=3x²+8x-3/6(x-1/3)(x-3/2)=3x²+8x-3/(3x-1)(2x-3)
deltaLicznika=64-4*3*(-3)=100
√deltaLicznika=10
x1=(-8-10)/6=-3
x2=(-8+10)/6=1/3
3x²+8x-3/6x²-11x+3=3x²+8x-3/6(x-1/3)(x-3/2)=3x²+8x-3/(3x-1)(2x-3)=3(x+3)(x-1/3)/(3x-1)(2x-3)=(x+3)(3x-1)/(3x-1)(2x-3)=(x+3)/(2x-3)
e)3x⁴-17x³+10x²/6x²-34x+20
6x²-34x+20≠0
3x²-17x+10≠0
delta=289-4*3*10=289-120=169
√delta=13
x1=(17-13)/6=2/3
x2=(17+13)/6=5
zatem x≠2/3 , x≠5
3x⁴-17x³+10x²/6x²-34x+20=3x⁴-17x³+10x²/6(x-2/3)(x-5)=3x⁴-17x³+10x²/2(3x-2)(x-5)=x²(3x²-17x+10)/2(3x-2)(x-5)=x²(3x²-17x+10)/2(3x-2)(x-5)=x²(3x-2)(x-5)/2(3x-2)(x-5)=x²/2
f)4x³-9x/4x³+12x²+9x
4x³+12x²+9x≠0
delta=144-4*4*9=0
x=-12/8=-3/2
zatem x≠-3/2
4x³-9x/4x³+12x²+9x=x(4x²-9)/4(x-3/2)(x-3/2)=x(4x²-9)/(2x-3)²=x(2x-3)(2x+3)/(2x-3)²=x(2x+3)/(2x-3)
g)x³+27/2x²-6x+18
2x²-6x+18≈0
x²-3x+9≈0
delta=9-4*1*9<0
x∉R
x³+27/2x²-6x+18=(x+3)(x²-3x+9)/2(x²-3x+9)=(x+3)/2
h)x⁵-x³-27x²+27/x⁴-3x³-x²+3x
x⁴-3x³-x²+3x≠0
x³(x-3)-x(x-3)=(x³-x)(x-3)=x(x²-1)(x-3)=x(x-1)(x+1)(x-3)
zatem x≠0,x≠1,x≠-1,x≠3
x⁵-x³-27x²+27/x⁴-3x³-x²+3x=x⁵-x³-27x²+27/x(x-1)(x+1)(x-3)=[x³(x²-1)-27(x²-1)]/x(x-1)(x+1)(x-3)=(x²-1)(x³-27)/x(x-1)(x+1)(x-3)=(x-1)(x+1)(x-3)(x²+3x+9)/x(x-1)(x+1)(x-3)=(x²+3x+9)/x
i)x³+x²-2x/x⁴+8x
x⁴+8x≠0
x(x³+8)≠0
x(x+2)(x²-2x+4)
zatem x≠0, x≠-2
x³+x²-2x/x⁴+8x=x³+x²-2x/x(x+2)(x²-2x+4)=x(x²+x-2)/x(x+2)(x²-2x+4)=(x²+x-2)/(x+2)(x²-2x+4)