Matematicas Algebra

Formulario Matemáticas Álgebra
1.Conjuntos Numéricos

IN = { ,2,3,4,...}
1
IN 0 = {0,1,2,3,4,...} = IN ∪ {0}
Z = {...,−4,−3−,2,−1,0,1,2,3,4,...}
a

Q =  ; a, b ∈ Z ∧ b ≠ 0
b


{

}

Q* = ...,± 3 ,± 2 ,±π ,± π ,...
IR = Q ∪ Q *

{

}

C = (a, b ) ∈ IR × IR / (a, b ) = a + bi; i = − 1
0

IN ⊂ IN ⊂ Z ⊂ Q ⊂ IR ⊂ C
Q* ⊂ IR
Q ∩ Q* = φ
Q ∪ Q* = IR

2.Potencias
b n= b ⋅ 444 b ⋅ b ⋅ b ⋅ b......b
⋅2
1b ⋅ b ⋅ b4 4444
3
n veces b
p

q

a ⋅a = a

p+q

a p ⋅ b p = (a ⋅ b )

p

a n ÷ a m = a n−m ⇔
p

an
= a n− m
m
a

a p ÷ b p = (a ÷ b ) ⇔

(a )

pq

ap a
= 
bp b

p

= a p ⋅q
p

1
1
a−p =   = p ⇔ a ≠ 0
a
a
−p

p

bp
a
b
  =   = p ⇔ a ≠ 0∧b ≠ 0
a
b
a
a 0 = 1 ⇔ a ≠ 0 → 0 0 ∉ IR

1/103.Raíces
x = n c ⇔ xn = c ∧ n ≠ 0
q
p

p

a =a

n

a ⋅ n b = n a ⋅b

n

a ÷ n b = n a ÷b ⇔

q

n
n

a na
=
b
b

a ⋅ n b = n an ⋅b ⇔ n an ⋅b = a ⋅ n b
pq

p⋅q

a= a
Racionalización :
a
b ab
⋅
=
b
bb

→
→
→

a
n

bm

n

⋅

n

b n −m
b n −m

=

a n b n−m
b

(

a
bm c a bm c
⋅
=
b−c
b± c bm c

)

4.Álgebra
ProductosNotables
(a ± b )2 = a 2 ± 2ab + b 2

(a ± b )3 = a 3 ± 3a 2 b + 3ab 2 ± b 3
(a + b )(a − b ) = a 2 − b 2
(x + a )(x + b ) = x 2 + x(a + b ) + ab
(a + b + c ) = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc
a 3 ± b 3 = (a ± b )(a 2 m bc + b 2 )

(a + b )n = ∑   ⋅ a n−i b i
i 
n

n

i=0



2/10

Triangulo

de

(a + b )0
(a + b )1
(a + b )2
(a + b )3
(a + b )4
(a + b )5
(a + b)6
(a + b )7
(a + b )8
(a + b )9
(a + b )10

Pascal
1
1
1
1
1
1
1
1
1
1

1

8
9

10

36

4

20

56

Ecuación de 1er grado
c−b
a
x + b = a + b
x − b = a − b


x = a ⇔ b ⋅ x = b ⋅ a ∧ b ≠ 0

x = a ∧b ≠ 0
b b


ax + b = c ⇒ x =

5.Sistemas de Ecuaciones de 1er grado
ax + by = c L(1)
dx + ey = f L(2)
Eliminación por sustitución
c − by
(1) ⇒x =
a
 c − by 
en (2) ⇒ d 
 + ey = f LLL
a

3/10

1
6

21
56

126
252
M

N

5

35

126

1

15

70

210

1

10

35

84
120

3

10

21

1

6

15

28

45

3

5

7

2

4

6

1

7
28

84
210

1
1
8
36

120

1
9

45

1
10

1
O

Eliminación por reducción
ax + by = c /⋅ −d
dx + ey = f / ⋅ a
⇒

−adx − bdy = −cd
+ adx + aey = + af

(1) + (2) ⇒ y (ae − bd ) = af − cd LLL
Eliminación por igualación
ax + by = c (1)
dx + ey = f (2)

c − by
a
f − dy
(2) ⇒ x =
c
c − by f − dy
⇒
=
LLL
a
c
(1) ⇒ x =

Regla de Cramer
ax + by = c
dx + ey = f
ce − bf
ae − bd
af − cd
y=
ae − bd
→ ae − bd ≠ 0 ⇒ ∃ ! solución
x=

sin .solución ⇔ ce − bf ≠ 0 ∨ af − cd ≠ 0
→ ae − bd = 0 ⇒no ∃ ! solución 
∞.soluciones ⇔ ce − bf = 0 ∨ af − cd = 0

4/10

6.Razones y proporciones
a
=k
b
ac

=

bd
 ⇔ a⋅d = b⋅c
a ÷b = c ÷d

d c
b = a

b = d
a c

a = b
ac
c d
Si = ⇒ 
bd
a + b = c + d ∨ a + b = c + d
a
c
d
d
a − b c − d a − b c − d

=
∨
=
c
d
d
a
a + b c + d
=

a −b c − d
a
P.Directa ⇒ = K
b
P.Inversa ⇒ a ⋅ b = K7.Inecuaciones
a⋅x +b

≥

x + b < a + b
x − b < x − b

 x ⋅ b < a ⋅ b


⇔b>0
x < a

b b


x a ⋅ b

x a
⇔b


b b
n
n
 x < a ⇔ n > 0 ∧ x, a > 0
n
 x > a n ⇔ n > 0 ∧ n. par ∧ x, a < 0

Intervalos:
Cerrado: [a, b] = {x ∈ IR / a ≤ x ≤ b}
Abierto: (a, b ) = ]a, b[ = {x ∈ IR / a < x < b}

5/10

Semiabierto o semicerrado:

[a, b[ = [a, b ) ={x ∈ IR / a ≤ x < b}
]a, b] = (a, b] = {x ∈ IR / a < x ≤ b}
por la derecha:

[a,+∞[ = [a,+∞ ) = {x ∈ IR / a ≤ x}
]a,+∞[ = (a,+∞ ) = {x ∈ IR / a < x}

por la izquierda:

Intervalos indeterminados:

]− ∞, a ] = (− ∞, a] = {x ∈ IR / x ≤ a}
]− ∞, a ] = (− ∞, a ) = {x ∈ IR / x < a}

8.Teoría de Conjuntos
A ∪ B = {x / x ∈ A ∨ x ∈ B}
A ∩ B = {x / x ∈ A ∧ x ∈ B}
A − B = {x / x ∈ A ∧ x ∉...