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Algebraické výrazy

1. Algebraické výrazy a jejich úpravy

1)

1xx+2xx+2+1

1xx+2xx+2+1=x+2xx+2x+x+2x+2=2x+22x+2x+2=2x+22(x+1)x+2=2x+2× x+22(x+1)=1x+1x1,x2

2)

(x3y2+x2y+x+y)÷(x2y2y2x2)


(x3y2+x2y+x+y)÷(x2y2y2x2)=x3+x2y+xy2+y3y2÷x4y4x2y2=x2(x+y)+y2(x+y)y2×x2y2(x2y2)(x2+y2)=(x2+y2)(x+y)y2×x2y2(x2y2)(x2+y2)=x2(x+y)(xy)(x+y)=x2xyx0,y0,x+1,y1

3)

[(n+2n2)3÷n3+4n2+4n3n212n+12]×n3


[(n+2n2)3÷n3+4n2+4n3n212n+12]×n3= [(n+2)3(n23÷n(n2+4n+4)3(n24n+4)]×n3= [(n+2)3(n23÷n(n+2)23(n2)2]×n3= (n+2)3(n2)3×3(n2)2n(n+2)2×n3=n+2n2 n0,n+2,n2

4)

a4b4a2b2÷[(1+b2a2)×(12ab+a2b2)]


a4b4a2b2÷[(1+b2a2)×(12ab+a2b2)]= (a2b2)(a2+b2)a2b2÷[a2+b2a2×b22ab+a2b2]= (a2b2)(a2+b2)a2b2÷[(a2+b2)×(ab)2a2b2]= (a2b2)(a2+b2)a2b2×a2b2(a2+b2)×(ab)2= (a2b2)(ab)2=(ab)(a+b)(ab)(ab)=a+babab,a0,b0

5)

2a(2a3a+1a2+32a22a+122a)×a3+1a22

2a(2a3a+1a2+32(a21)a+12(1a))×a3+1a(a1)=

2a(2a3a+1a2+32(a1)(a+1)a+12(1a))×a3+1a(a1)=

2a(2(a1)(2a3)a23+(a+1)(a+1)2(a+1)(a1))×a3+1a(a1)=

2a(2(a23a2a+3)a23+a2+2a+12(a+1)(a1))×a3+1a(a1)=

2a(4a210a+62+2a2(a+1)(a1))×a3+1a(a1)=

2a4a28a+42(a+1)(a1)×a3+1a(a1)=

2a4(a22a+1)2(a+1)(a1)×a3+1a(a1)=

2a4(a1)22(a+1)(a1)×a3+1a(a1)=

2a2(a3+1)a(a+1)=2a2(a+1)2(a3+1)a(a+1)=

2a3+2a22a32a(a+1)=2a22a(a+1)=

2(a+1)(a1)a(a+1)=2(a1)a


x1

x0

6)

(1a+12aa21)×(1a1)


(1a+12aa21)×(1a1)= (a1)2a(a+1)(a1)×1aa= a12a(a+1)(a1)×(a1)a= a1(a+1)(a1)×(a1)a=(a+1)(a+1)(a1)×(a1)a=1aa0,a1,a+1

7)

a2+aba2+b2×(aabba+b)


a2+aba2+b2×(aabba+b)= a(a+b)a2+b2×(a(a+b)b(ab)(ab)(a+b))= a(a+b)a2+b2×(a2+abab+b2(ab)(a+b))= a(a+b)a2+b2×(a2+b2(ab)(a+b))=aaba+b,ab

8)

[3a8a3÷4a24(a2+2a+4)]×a24a+4a


[3a8a3÷4a24(a2+2a+4)]×a24a+4a= 3a(2a)(4+2a+a2)×4(a2+2a+4)(2a)(2+a)×a24a+4a= 12a(2a)(2a)(2+a)×(a2)2a=122+aa0,a+2,a2

9)

(4x28x+4x2+1÷x+13)÷6x6x41


(4x28x+4x2+1÷x+13)÷6x6x41= (2x2)2x2+1×3x+1×(x21)(x2+1)6(x1)= (2x2)2x2+1×3x+1×(x1)(x+1)(x2+1)6(x1)= 3(2x2)26=(2x2)22=4x28x+42=2(2x24x+22=2x24x+2=2(x1)2x+1,x1

10)

6a+(aa2aa+2)÷4aa42a3+8a16


6a+a(a+2)a(a2)(a2)(a+2)÷4aa3(a2)8(a2)= 6a+a2+2aa2+2a(a2)(a+2)×a3(a2)8(a2)4a= 6a+4a(a2)(a+2)×(a2)(a3+8)4a= 6a+(a2)(a+2)(a22a+4)(a2)(a+2)=6a+a22a+4=a2+4a+4=(a+2)2a0,a+2,a2