מתמטיקה - 4 יח"ל - כיתה י"ב - חלק א'

ﺟﺑﺭ ﻭﻣﻘﺩّﻣﺔ ﻟﻠﺗّﺣﻠﻳﻝ ﺍﻟﺭّﻳﺎﺿﻲ ﻟﻠﺩّﻭﺍﻝ ﺍﻷﺳّﻳّﺔ ﻭﺍﻟﻠّﻭﻏﺭﻳﺛﻣﻳّﺔ

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ﺍﻟﻔﺻﻝ :1 ﻗﻭﻯ ﻭﺟﺫﻭﺭ .

ﺟﺒﺮ وﻣﻘﺪّﻣﺔ ﻟﻠﺘّﺤﻠﯿﻞ اﻟﺮّﻳﺎﺿﻲ ﻟﻠﺪّوال اﻷﺳّﯿّﺔ واﻟﻠّﻮﻏﺮﻳﺜﻤﯿّﺔ

ﺍﻟﻔﺼﻞ :1 ﻗﻮﻯ ﻭﺟﺬﻭﺭ

ﺃ . ﻗﻮﺍﻧﻴﻦ ﺍﻟﻘﻮﻯ ﻓﻲ ﺍﻷُﺳُﺲ ﺍﻟﻄّﺒﻴﻌﻴّﺔ ﻭﺍﻷُﺳُﺲ ﺍﻟﺼّﺤﻴﺤﺔ

ﺗﺬﻛﻴﺮ : ﻓﻲ ﺗﺮﺗﻴﺐ ﺍﻟﻌﻤﻠﻴّﺎﺕ ﺍﻟﺤﺴﺎﺑﻴّﺔ :

ﺍﻟﻌﻤﻠﻴّﺎﺕ ﻓﻲ ﺍﻷﻗﻮﺍﺱ ﺗﺴﺒﻖ ﻋﻤﻠﻴّﺎﺕ ﺍﻟﻘﻮﻯ / ﺍﻟﺠﺬﻭﺭ . ﻋﻤﻠﻴّﺎﺕ ﺍﻟﻘﻮﻯ / ﺍﻟﺠﺬﻭﺭ ﺗﺴﺒﻖ ﻋﻤﻠﻴّﺎﺕ ﺍﻟﻀّﺮﺏ / ﺍﻟﻘﺴﻤﺔ . ﻋﻤﻠﻴّﺎﺕ ﺍﻟﻀّﺮﺏ / ﺍﻟﻘﺴﻤﺔ ﺗﺴﺒﻖ ﻋﻤﻠﻴّﺎﺕ ﺍﻟﺠﻤﻊ / ﺍﻟﻄّﺮﺡ .

ﺳﻨﻘﺪّﻡ ﺗﻌﺮﻳﻔﻴﻦ ﻳﺘﻌﻠّﻘﺎﻥ ﺑﺎﻟﻘﻮﻯ، ﻭﻣﻦ ﺛﻢ ﻗﻮﺍﻧﻴﻦ ﺍﻟﻘﻮﻯ .

n a a a a      

ّ ﻟﻠﻘﻮ ة n ھﻮ ﺣﺎﺻﻞ اﻟﻀّﺮب : 

   ﻟﻜﻞ ﻋﺪد ﺣﻘﯿﻘﻲ a وﻋﺪد طﺒﯿﻌﻲ n ﻧﻌﺮّف : a

 

n ﻣﺮّات

ّة أُس اﻟﻘﻮ .

ّة أﺳﺎس اﻟﻘﻮ ، وﻧﺴﻤّﻲ اﻟﻌﺪد n

ّ ﻧﺴﻤ ﻲ اﻟﻌﺪد a

 

3  ( ) ( ) ( ) ( ) 2

2 2 2 8       

  ﻣﺜﺎل:

n  

0 a  وَ

a  ﻳﺘﺤﻘّﻖ :

1

) 0

0 ﻟﯿﺲ ﻣﻌﺮّﻓًﺎ . (

   ﻟﻜﻞ ﻋﺪد ﺣﻘﯿﻘﻲ

0

1

a

n

a

ﻗﻮاﻧﯿﻦ اﻟﻘﻮى :

b  (

a  وَ 0

) ﻓﻲ اﻟﻘﻮاﻧﯿﻦ اﻟﻤﺨﺘﻠﻔﺔ m وَ n ھﻤﺎ ﻋﺪدان ﺻﺤﯿﺤﺎ ن ، 0







 0

1

1

n



1





a

a

n

a

a



 m n m n a a a  

m



a

m n







 





a

n

a

 n n n a b a b  _ i

a a _ i  

 b b a k  n a a

n

 n m m n





 

n

 n b  a k a k a b a

n

a a  _ i )ﻟﻜﻞّ m ﻭَ n

n m mn

ﱠﻴﻴْﻦ ﻁﺒﻴﻌ .(

ﻋﻠﻰ ﺳﺒﻴﻞ ﺍﻟﻤﺜﺎﻝ، ﻧُﺒﻴّﻦ ﺃﻧّﻪ ﻳﺘﺤﻘّﻖ :

m _ i _ i a a

 





 

   



  

 

  



n m m m m m m

n      ohngq ohngq ohngq ohngq ﻣﺮّﺍﺕ ﻣﺮّﺍﺕ ﻣﺮّﺍﺕ ﻣﺮّﺍﺕ m m m a a a a ...a (a a...a)(a a...a)...(a a...a)

ohngq ﻣﺮّﺍﺕ

n

a a ...    



n

m n

a

a

ohn  gq ﻣﺮّﺍﺕ m n 

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