No cálculo algébrico, algumas expressões representadas por produtos de expressões algébricas , aparecem com muita frequência. Pela importância que representam no cálculo algébrico , essas expressões são denominadas Produtos Notáveis e são utilizados principalmente para a fatoração de polinômios e evitar erros com sinais.[ 1]
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
.
{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}.}
Regra básica : Quadrado do primeiro termo, somado ao dobro do primeiro termo multiplicado pelo segundo termo, somado ao quadrado do segundo termo.[ 2]
Prova :
(
a
+
b
)
2
=
(
a
+
b
)
⋅
(
a
+
b
)
=
a
⋅
(
a
+
b
)
+
b
⋅
(
a
+
b
)
=
a
2
+
a
b
+
a
b
+
b
2
=
a
2
+
2
a
b
+
b
2
{\displaystyle (a+b)^{2}=(a+b)\cdot (a+b)=a\cdot (a+b)+b\cdot (a+b)=a^{2}+ab+ab+b^{2}=a^{2}+2ab+b^{2}}
Exemplo:
(
4
x
5
y
+
z
)
2
=
(
4
x
5
y
+
z
)
⋅
(
4
x
5
y
+
z
)
=
4
x
5
y
⋅
(
4
x
5
y
+
z
)
+
z
⋅
(
4
x
5
y
+
z
)
=
16
x
2
25
y
2
+
4
x
z
5
y
+
4
x
z
5
y
+
z
2
=
16
x
2
25
y
2
+
8
x
z
5
y
+
z
2
{\displaystyle \left({\dfrac {4x}{5y}}+z\right)^{2}=\left({\dfrac {4x}{5y}}+z\right)\cdot \left({\dfrac {4x}{5y}}+z\right)={\dfrac {4x}{5y}}\cdot \left({\dfrac {4x}{5y}}+z\right)+z\cdot \left({\dfrac {4x}{5y}}+z\right)={\dfrac {16x^{2}}{25y^{2}}}+{\dfrac {4xz}{5y}}+{\dfrac {4xz}{5y}}+z^{2}={\dfrac {16x^{2}}{25y^{2}}}+{\frac {8xz}{5y}}+z^{2}}
(
8
x
+
a
)
2
=
(
8
x
+
a
)
⋅
(
8
x
+
a
)
=
8
x
⋅
(
8
x
+
a
)
+
a
⋅
(
8
x
+
a
)
=
64
x
2
+
8
a
x
+
8
a
x
+
a
2
=
64
x
2
+
16
a
x
+
a
2
{\displaystyle (8x+a)^{2}=(8x+a)\cdot (8x+a)=8x\cdot (8x+a)+a\cdot (8x+a)=64x^{2}+8ax+8ax+a^{2}=64x^{2}+16ax+a^{2}}
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
{\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}
Regra básica : Quadrado do primeiro termo, subtraído o dobro do produto do primeiro termo pelo segundo termo, somado ao quadrado do segundo termo.
Prova :
(
a
−
b
)
2
=
(
a
−
b
)
⋅
(
a
−
b
)
=
a
⋅
(
a
−
b
)
−
b
⋅
(
a
−
b
)
=
a
2
−
a
b
−
a
b
+
b
2
=
a
2
−
2
a
b
+
b
2
{\displaystyle (a-b)^{2}=(a-b)\cdot (a-b)=a\cdot (a-b)-b\cdot (a-b)=a^{2}-ab-ab+b^{2}=a^{2}-2ab+b^{2}}
Exemplos :
(
3
m
4
n
−
p
)
2
=
9
m
2
16
n
2
−
3
m
p
2
n
+
p
2
{\displaystyle \left({\frac {3m}{4n}}-p\right)^{2}={\frac {9m^{2}}{16n^{2}}}-{\frac {3mp}{2n}}+p^{2}}
(
1
−
2
x
)
2
=
1
−
4
x
+
4
x
2
{\displaystyle (1-2x)^{2}=1-4x+4x^{2}}
(
a
+
b
)
⋅
(
a
−
b
)
=
a
2
−
b
2
{\displaystyle (a+b)\cdot (a-b)=a^{2}-b^{2}}
Regra básica : Quadrado do primeiro termo subtraído o quadrado do segundo termo.
Prova :
(
a
+
b
)
⋅
(
a
−
b
)
=
a
2
−
a
b
+
a
b
−
b
2
=
a
2
+
a
b
⋅
(
1
−
1
)
−
b
2
=
a
2
−
b
2
{\displaystyle (a+b)\cdot (a-b)=a^{2}-ab+ab-b^{2}=a^{2}+ab\cdot (1-1)-b^{2}=a^{2}-b^{2}}
Exemplos :
(
a
2
+
b
3
)
⋅
(
a
2
−
b
3
)
=
a
4
−
a
2
b
3
+
a
2
b
3
−
b
6
=
a
4
+
a
2
b
3
⋅
(
1
−
1
)
−
b
6
=
a
4
−
b
6
{\displaystyle (a^{2}+b^{3})\cdot (a^{2}-b^{3})=a^{4}-a^{2}b^{3}+a^{2}b^{3}-b^{6}=a^{4}+a^{2}b^{3}\cdot (1-1)-b^{6}=a^{4}-b^{6}}
(
a
x
+
2
)
.
(
a
x
−
2
)
=
a
2
x
2
−
2
a
x
+
2
a
x
−
4
=
a
2
x
2
+
2
a
⋅
(
1
−
1
)
x
−
4
=
a
2
x
2
−
4
{\displaystyle \left({\frac {a}{x}}+2\right).\left({\frac {a}{x}}-2\right)={\frac {a^{2}}{x^{2}}}-{\frac {2a}{x}}+{\frac {2a}{x}}-4={\frac {a^{2}}{x^{2}}}+{\frac {2a\cdot (1-1)}{x}}-4={\frac {a^{2}}{x^{2}}}-4}
Decomposição volumétrica do binômio ao cubo
(
x
+
y
)
3
=
x
3
+
3
x
2
y
+
3
x
y
2
+
y
3
{\displaystyle (x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}}
Regra básica : O cubo do primeiro termo, somado o triplo do produto do quadrado do primeiro termo pelo segundo termo, somado ao triplo do produto do primeiro termo pelo quadrado do segundo termo, somado ao cubo do segundo termo.
Exemplos :
(
m
+
3
n
)
3
=
m
3
+
3
⋅
m
2
⋅
3
n
+
3
⋅
m
⋅
(
3
n
)
2
+
(
3
n
)
3
=
m
3
+
9
m
2
n
+
27
m
n
2
+
27
n
3
{\displaystyle (m+3n)^{3}=m^{3}+3\cdot m^{2}\cdot 3n+3\cdot m\cdot (3n)^{2}+(3n)^{3}=m^{3}+9m^{2}n+27mn^{2}+27n^{3}}
(
x
+
2
)
3
=
x
3
+
6
x
2
+
12
x
+
8
{\displaystyle (x+2)^{3}=x^{3}+6x^{2}+12x+8}
(
a
+
3
b
)
3
=
a
3
+
9
a
2
b
+
27
a
b
2
+
27
b
3
{\displaystyle (a+3b)^{3}=a^{3}+9a^{2}b+27ab^{2}+27b^{3}}
(
a
−
b
)
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
{\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}}
Regra básica : Para calcular o cubo da diferença faça: O cubo do 1° termo, subtraído o triplo do produto do quadrado do 1° termo pelo segundo termo, somado ao triplo do produto do 1° termo pelo quadrado do 2° termo, subtraído o cubo do 2° termo.
(
a
−
b
)
3
=
(
a
−
b
)
⋅
(
a
−
b
)
2
=
(
a
−
b
)
⋅
(
a
2
−
2
a
b
+
b
2
)
=
a
3
−
2
a
2
b
+
a
b
2
−
a
2
b
+
2
a
b
2
−
b
3
=
a
3
−
3
a
2
b
+
3
a
b
2
−
b
3
{\displaystyle (a-b)^{3}=(a-b)\cdot (a-b)^{2}=(a-b)\cdot (a^{2}-2ab+b^{2})=a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}}
Exemplos :
(
b
−
2
c
)
3
=
b
3
−
6
b
2
c
+
12
b
c
2
−
8
c
3
{\displaystyle (b-2c)^{3}=b^{3}-6b^{2}c+12bc^{2}-8c^{3}}
(
x
y
−
a
b
)
3
=
x
3
y
3
−
3
a
x
2
b
y
2
+
3
a
2
x
b
2
y
−
a
3
b
3
{\displaystyle \left({\frac {x}{y}}-{\frac {a}{b}}\right)^{3}={\frac {x^{3}}{y^{3}}}-{\frac {3ax^{2}}{by^{2}}}+{\frac {3a^{2}x}{b^{2}y}}-{\frac {a^{3}}{b^{3}}}}
(
1
−
x
)
3
=
1
−
3
x
+
3
x
2
−
x
3
{\displaystyle (1-x)^{3}=1-3x+3x^{2}-x^{3}}
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
a
b
+
2
a
c
+
2
b
c
{\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc}
(
a
+
b
+
c
)
2
=
a
2
+
a
b
+
a
c
+
b
2
+
a
b
+
b
c
+
a
c
+
b
c
+
c
2
{\displaystyle (a+b+c)^{2}=a^{2}+ab+ac+b^{2}+ab+bc+ac+bc+c^{2}}
→
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
a
b
+
2
a
c
+
2
b
c
{\displaystyle \rightarrow (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc}
Exemplos :
(
x
+
y
+
z
)
2
=
x
2
+
y
2
+
z
2
+
2
x
y
+
2
x
z
+
2
y
z
{\displaystyle (x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2xz+2yz}
(
x
−
2
y
−
3
)
2
=
x
2
+
(
−
2
y
)
2
+
(
−
3
)
2
+
2
x
(
−
2
y
)
+
2
x
(
−
3
)
+
2
(
−
2
y
)
(
−
3
)
{\displaystyle (x-2y-3)^{2}=x^{2}+(-2y)^{2}+(-3)^{2}+2x(-2y)+2x(-3)+2(-2y)(-3)}
(
5
x
+
4
y
+
15
z
)
2
=
25
x
2
+
16
y
2
+
225
z
2
+
40
x
y
+
75
x
z
+
60
y
z
{\displaystyle (5x+4y+15z)^{2}=25x^{2}+16y^{2}+225z^{2}+40xy+75xz+60yz}
(
x
+
a
)
⋅
(
x
+
b
)
=
x
2
+
(
a
+
b
)
x
+
a
b
{\displaystyle (x+a)\cdot (x+b)=x^{2}+(a+b)x+ab}
(
x
+
a
)
⋅
(
x
+
b
)
=
x
⋅
(
x
+
b
)
+
a
⋅
(
x
+
b
)
=
x
2
+
b
x
+
a
x
+
a
b
⇒
x
2
+
(
a
+
b
)
x
+
a
b
{\textstyle (x+a)\cdot (x+b)=x\cdot (x+b)+a\cdot (x+b)=x^{2}+bx+ax+ab\Rightarrow x^{2}+(a+b)x+ab}
Exemplos :
(
x
+
4
)
⋅
(
x
+
3
)
=
x
2
+
(
4
+
3
)
⋅
x
+
4
⋅
3
=
x
2
+
7
x
+
12
{\displaystyle (x+4)\cdot (x+3)=x^{2}+(4+3)\cdot x+4\cdot 3=x^{2}+7x+12}
(
x
−
2
)
⋅
(
x
−
6
)
=
x
2
+
(
−
2
−
6
)
⋅
x
+
(
−
2
)
⋅
(
−
6
)
=
x
2
−
8
x
+
12
{\displaystyle (x-2)\cdot (x-6)=x^{2}+(-2-6)\cdot x+(-2)\cdot (-6)=x^{2}-8x+12}
(
x
−
1
)
⋅
(
x
+
5
)
=
x
2
+
(
−
1
+
5
)
⋅
x
+
5
⋅
(
−
1
)
=
x
2
+
4
x
−
5
{\displaystyle (x-1)\cdot (x+5)=x^{2}+(-1+5)\cdot x+5\cdot (-1)=x^{2}+4x-5}
equações polinomiais .
Assumindo uma equação polinomial de grau 2 podemos escrevê-la como:
a
x
2
+
b
x
+
c
=
0
{\displaystyle ax^{2}+bx+c=0}
x
2
−
x
⋅
(
x
1
+
x
2
)
+
x
1
x
2
=
0
{\displaystyle x^{2}-x\cdot (x_{1}+x_{2})+x_{1}x_{2}=0}
Onde a segunda pode ser fatorada como
(
x
−
x
1
)
⋅
(
x
−
x
2
)
=
0
{\displaystyle (x-x_{1})\cdot (x-x_{2})=0}
a
⋅
(
x
−
x
1
)
⋅
(
x
−
x
2
)
=
0
{\displaystyle a\cdot (x-x_{1})\cdot (x-x_{2})=0}
(
a
+
b
)
⋅
(
a
2
−
a
b
+
b
2
)
=
a
3
+
b
3
{\displaystyle (a+b)\cdot (a^{2}-ab+b^{2})=a^{3}+b^{3}}
Prova : Considerando
(
a
+
b
)
⋅
(
a
2
−
a
b
+
b
2
)
,
{\displaystyle (a+b)\cdot (a^{2}-ab+b^{2}),}
(
a
+
b
)
⋅
(
a
2
−
a
b
+
b
2
)
=
a
3
−
a
2
b
+
a
b
2
+
a
2
b
−
a
b
2
+
b
3
=
a
3
+
b
3
.
{\displaystyle (a+b)\cdot (a^{2}-ab+b^{2})=a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}=a^{3}+b^{3}.}
(
x
+
5
)
⋅
(
x
2
−
5
x
+
25
)
=
x
3
+
5
3
=
x
3
+
125
{\displaystyle (x+5)\cdot (x^{2}-5x+25)=x^{3}+5^{3}=x^{3}+125}
(
2
x
+
3
)
⋅
(
4
x
2
−
6
x
+
9
)
=
(
2
x
)
3
+
3
3
=
8
x
3
+
27
{\displaystyle (2x+3)\cdot (4x^{2}-6x+9)=(2x)^{3}+3^{3}=8x^{3}+27}
(
a
−
b
)
⋅
(
a
2
+
a
b
+
b
2
)
=
a
3
−
b
3
{\displaystyle (a-b)\cdot (a^{2}+ab+b^{2})=a^{3}-b^{3}}
Prova : Considerando
(
a
−
b
)
⋅
(
a
2
+
a
b
+
b
2
)
,
{\displaystyle (a-b)\cdot (a^{2}+ab+b^{2}),}
(
a
−
b
)
⋅
(
a
2
+
a
b
+
b
2
)
=
a
3
+
a
2
b
−
a
b
2
−
a
2
b
+
a
b
2
−
b
3
=
a
3
−
b
3
{\displaystyle (a-b)\cdot (a^{2}+ab+b^{2})=a^{3}+a^{2}b-ab^{2}-a^{2}b+ab^{2}-b^{3}=a^{3}-b^{3}}
Exemplo
(
x
−
3
)
⋅
(
x
2
+
3
x
+
9
)
=
x
3
−
3
3
=
x
3
−
27
{\displaystyle (x-3)\cdot (x^{2}+3x+9)=x^{3}-3^{3}=x^{3}-27}
Notas e referências Equação polinomial
Referências
