PRODUTOS NOTÁVEIS E FATORAÇÃO
Autor: TURMA MARANATA IFMT | Publicado: 2011-10-08T11:06:00-07:00

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 <![endif]-->  &nbsp;Produtos Notáveis&nbsp;&nbsp;

 são aqueles produtos que são freqüentemente usados e para evitar a multiplicação de termo a termo, existem algumas fórmulas que convém serem memorizadas.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1) Soma pela diferença: quadrado do primeiro menos o quadrado do segundo.
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ( a + b ).( a – b ) = a² - b² 
&nbsp;2) Quadrado da soma: quadrado do primeiro, mais duas vezes o primeiro pelo segundo, mais o quadrado do segundo. 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ( a + b )² = a² + 2ab +b² 
&nbsp;3) Quadrado da diferença: quadrado do primeiro, menos duas vezes o primeiro pelo segundo, mais o quadrado do segundo. 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ( a – b )² = a² - 2ab + b²

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 



Fatoração&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Fatorar é transformar equações algébricas em produtos de duas ou mais expressões, chamadas fatores.
Ex: ax + ay = a.(x+y)
Existem vários casos de fatoração como:
1) Fator Comum em evidência&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Quando os termos apresentam fatores comuns
Observe o polinômio:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ax + ay &nbsp;» Ambos os termos apresentam o fator a em evidência.
Assim: ax + ay = a.(x+y) 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Forma fatorada&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2) Fatoração por agrupamento
Consiste em aplicar duas vezes o caso do fator comum em alguns polinômios especiais.
Como por exemplo:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ax + ay + bx + by
Os dois primeiros termos possuem em comum o fator a , os dois últimos termos possuem em comum o fator b. Colocando esses termos em evidência:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a.(x+y) + b.(x+y)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Este novo polinômio possui o termo (x+y) em comum. Assim colocando-o em evidência:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x+y).(a+b)&nbsp;&nbsp; Ou seja: &nbsp;ax + ay + bx + by = (x+y).(a+b)&nbsp;&nbsp;&nbsp;&nbsp;










 
¨É possível reconhecer a utilidade de uma idéia, sem contudo compreender como usá-la&nbsp; adequadamente.¨&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Joahnn Goethe ( 1749- 1832 ) Romancista, poeta e teatrólogo alemão.

EXERCÍCIOS FENIX 
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   UnhideWhenUsed="false" Name="Light Shading Accent 5"/>   <w:LsdException Locked="false" Priority="61" SemiHidden="false"
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   UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/>   <w:LsdException Locked="false" Priority="66" SemiHidden="false"
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   UnhideWhenUsed="false" QFormat="true" Name="Book Title"/>         <![endif]--><!--[if gte mso 10]> 
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 <![endif]-->  
      01)   Fatore a expressão E = a2 + ba + 2a + 2b  
   
   
  
      a)      E = (a   + b) (a + 2)  
      b)      E = (a   + b) (a - 2)  
      c)      E = (a   - b) (a - b)  
      d)      E = (a   + b) (a + b)  
      e)      E = (a   + 2b) (a - 2)  
   
  


      02)   Fatore a expressão: 3xy + 3 - x - 9y   
   
   
  
      a)      (x - 3)   . (3y - 1)  
      b)      (x - 3)   . (x + 3)  
      c)      (x + 9)   . (x - 9)  
      d)      (x - 3)   . (9y + 1)  
      e)      (x - 3)   . (3y + 1)  
   
  


      03)   Fatore a expressão: abd - abe + acd - ace  
   
   
  
      a)      a2   . (d + e) (b + c)  
      b)      a . (d   + e) (b - c)  
      c)      a . (d   - e) (b + c2)  
      d)      a . (d   - e) (b2 + c2)  
      e)      a . (d   - e) (b + c)  
   
  


      04)   Efetue o produto (x - y) . (x + y) . (x2 + y2).  
   
   
  
      a)      x4   - y4  
      b)      x4   + 4x2y2 + y4  
      c)      x4   - 2x2y2 + y4  
      d)      x4   + y4  
      e)      x4   + 2x2y2 + y4  
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 <![endif]--><!--[if gte mso 9]>           <![endif]--><!--[if gte mso 9]>     Normal   0   false      21         false   false   false      PT-BR   X-NONE   X-NONE                                                                                                     <![endif]--><!--[if gte mso 9]>  <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
  DefSemiHidden="true" DefQFormat="false" DefPriority="99"
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   UnhideWhenUsed="false" QFormat="true" Name="heading 1"/>                                                         <w:LsdException Locked="false" Priority="10" SemiHidden="false"
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   UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/>   <w:LsdException Locked="false" Priority="22" SemiHidden="false"
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   UnhideWhenUsed="false" Name="Colorful List Accent 1"/>   <w:LsdException Locked="false" Priority="73" SemiHidden="false"
   UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/>   <w:LsdException Locked="false" Priority="60" SemiHidden="false"
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   UnhideWhenUsed="false" Name="Light Grid Accent 2"/>   <w:LsdException Locked="false" Priority="63" SemiHidden="false"
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   UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/>   <w:LsdException Locked="false" Priority="60" SemiHidden="false"
   UnhideWhenUsed="false" Name="Light Shading Accent 4"/>   <w:LsdException Locked="false" Priority="61" SemiHidden="false"
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