Formulario de álgebra
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PROPIEDADES ARITMÉTICAS
ASOCIATIVA
𝑎(𝑏𝑐) = (𝑎𝑏)𝑐
CONMUTATIVA 𝑎 + 𝑏 = 𝑏 + 𝑎 𝑦 𝑎𝑏 = 𝑏𝑎
DISTRIBUTIVA
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
LEY DE SIGNOS
MULTIPLICACIÓN
DIVISIÓN
(+) × (+) = (+)
(+) ÷ (+) = (+)
(−) × (−) = (+)
(−) ÷ (−) = (+)
(+) × (−) = (−)
(+) ÷ (−) = (−)
(−) × (+) = (−)
(−) ÷ (+) = (−)
EJEMPLOS DE OPERACIONES ARITMÉTICAS
𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐)
𝑏
𝑎𝑏
𝑎( ) =
𝑐
𝑐
𝑎
( )
𝑏 = 𝑎
𝑐
𝑏𝑐
𝑎
𝑎𝑐
=
𝑏
𝑏
(𝑐 )
𝑎 𝑐 𝑎𝑑 − 𝑏𝑐
− =
𝑏 𝑑
𝑏𝑑
𝑎−𝑏 𝑏−𝑎
=
𝑐−𝑑 𝑑−𝑐
𝑎+𝑏 𝑎 𝑏
= +
𝑐
𝑐 𝑐
𝑎𝑏 + 𝑎𝑐
= 𝑏 + 𝑐, 𝑎 ≠ 0
𝑎
𝑎
( ) 𝑎𝑑
𝑏 =
𝑐
( ) 𝑏𝑐
𝑑
𝑎 𝑐 𝑎𝑑 + 𝑏𝑐
+ =
𝑏 𝑑
𝑏𝑑
ECUACIÓN CUADRÁTICA
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 → 𝑥 =
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
CURSO DE ÁLGEBRA
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√𝑎 = 𝑏 ↔ 𝑎 = 𝑏 𝑛
𝑚 𝑛
√ √𝑎 =
√𝑎 =
𝑛
𝑛
𝑎 √𝑎
√ =𝑛
𝑏
√𝑏
𝑛
𝑛
√𝑎𝑛 = 𝑎, 𝑠𝑖 𝑛 𝑒𝑠 𝑖𝑚𝑝𝑎𝑟
1
1
𝑎𝑛
𝑛
𝑛
√𝑎𝑏 = √𝑎 ⋅ √𝑏
√𝑎
𝑚
𝑎𝑥
√𝑎 𝑥
√ 𝑦=𝑚
𝑏
√𝑏 𝑦
𝑚
𝑛
𝑎−𝑚 =
1
; 𝑎≠0
𝑎𝑚
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
𝑚
𝑛
(𝑎𝑚 )𝑛 = 𝑎𝑚⋅𝑛 = 𝑎𝑛⋅𝑚 = (𝑎𝑛 )𝑚
𝑎 𝑛 = √𝑎𝑚
(𝑎𝑚 ⋅ 𝑏 𝑛 ⋅ 𝑐 𝑝 )𝑥 = 𝑎𝑚𝑥 ⋅ 𝑏 𝑛𝑥 ⋅ 𝑐 𝑝𝑥
𝑥
𝑎𝑚
𝑎𝑚⋅𝑥
𝑎 −𝑚
𝑏 𝑚
( 𝑛 ) = 𝑛⋅𝑥
( )
=( )
𝑏
𝑏
𝑏
𝑎
𝑎𝑚 ⋅ 𝑎𝑛 = 𝑎𝑚+𝑛
PRODUCTOS NOTABLES
(𝑎 +
𝑏)2
=
𝑎2
+ 2𝑎𝑏 + 𝑏 2
3
3
(𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2
2
(𝑎 + 𝑏) = 𝑎 + 3𝑎 𝑏 + 3𝑎𝑏 2 + 𝑏 3
(𝑎 + 𝑏)3 = 𝑎3 + 𝑏 3 + 3𝑎𝑏(𝑎 + 𝑏)
(𝑎 − 𝑏)3 = 𝑎3 − 3𝑎2 𝑏 + 3𝑎𝑏 2 − 𝑏 3
(𝑎 − 𝑏)3 = 𝑎3 − 𝑏 3 − 3𝑎𝑏(𝑎 − 𝑏)
𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
(𝑥 + 𝑎)(𝑥 + 𝑏) = 𝑥 2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏
(𝑎 + 𝑏)2 + (𝑎 − 𝑏)2 = 2(𝑎2 + 𝑏 2 )
(𝑎 + 𝑏)2 − (𝑎 − 𝑏)2 = 4𝑎𝑏
(𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) = 𝑎3 + 𝑏 3
(𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ) = 𝑎3 − 𝑏 3
(𝑎 + 𝑏 + 𝑐)2 = 𝑎2 + 𝑏 2 + 𝑐 2 + 2𝑎𝑏 + 2𝑏𝑐 + 2𝑎𝑐
(𝑎2 + 𝑎𝑏 + 𝑏 2 )(𝑎2 − 𝑎𝑏 + 𝑏 2 ) = 𝑎4 + 𝑎2 𝑏 2 + 𝑎4
(𝑎 + 𝑏 + 𝑐)3 = 𝑎3 + 𝑏 3 + 𝑐 3 + 3(𝑎 + 𝑏)(𝑎 + 𝑐)(𝑏 + 𝑐)
FACTORIZACIÓN
𝑎 + 2𝑎𝑚 𝑏𝑛 + 𝑏 2𝑛 = (𝑎𝑚 + 𝑏 𝑛 )2
𝑎2𝑚 − 2𝑎𝑚 𝑏𝑛 + 𝑏 2𝑛 = (𝑎𝑚 − 𝑏 𝑛 )2
𝑎2𝑚 − 𝑏 2𝑛 = (𝑎𝑚 + 𝑏 𝑛 )(𝑎𝑚 − 𝑏 𝑛 )
3𝑚
𝑎 + 𝑏 3𝑛 = (𝑎𝑚 + 𝑏 𝑛 )(𝑎2𝑚 − 𝑎𝑚 𝑏 𝑛 + 𝑏 2𝑛 )
𝑎3𝑚 − 𝑏 3𝑛 = (𝑎𝑚 − 𝑏 𝑛 )(𝑎2𝑚 + 𝑎𝑚 𝑏 𝑛 + 𝑏 2𝑛 )
𝑥 2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏)
𝑛
𝑚𝑛
𝑎0 = 1; 𝑎 ≠ 0
2𝑚
RADICALES
𝑛
LEYES DE EXPONENTES
√𝑎𝑛 = |𝑎|, 𝑠𝑖 𝑛 𝑒𝑠 𝑝𝑎𝑟
𝑎𝑥 2𝑚 + 𝑏𝑥 𝑚 𝑦 𝑛 + 𝑐𝑦 𝑛 = (𝑎1 𝑥 𝑚 + 𝑐1 𝑦 𝑛 )(𝑎2 𝑥 𝑚 + 𝑐2 𝑦 𝑛 )
𝑎1 𝑥 𝑚
𝑐1 𝑦 𝑛 ⇒ 𝑎2 𝑐1 𝑥 𝑚 𝑦 𝑛
(+)
𝑎2 𝑥 𝑚
𝑐2 𝑦 𝑛 ⇒ 𝑎1 𝑐2 𝑥 𝑚 𝑦 𝑛
𝑏𝑥 𝑚 𝑦 𝑛
DESIGUALDADES
𝑆𝑖 𝑎 < 𝑏 → 𝑎 + 𝑐 < 𝑏 + 𝑐 𝑦 𝑎 − 𝑐 < 𝑏 − 𝑐
𝑆𝑖 𝑎 < 𝑏 𝑦 𝑐 > 0 → 𝑎𝑐 < 𝑏𝑐 𝑦 𝑎/𝑐 < 𝑏/𝑐
𝑆𝑖 𝑎 < 𝑏 𝑦 𝑐 < 0 → 𝑎𝑐 > 𝑏𝑐 𝑦 𝑎/𝑐 > 𝑏/𝑐
Formulario de álgebra
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FACTORIAL Y NÚMERO COMBINATORIO
𝑛! = 1 × 2 × 3 × 4 × ⋯ × (𝑛 − 1) × 𝑛 ; 𝑛 ∈ ℕ; 𝑛 > 1
1! = 1
0! = 1
𝑛!
𝑛
𝐶𝑘𝑛 = ( ) =
𝑘
(𝑛 − 𝑘)! 𝑘!
𝑛
𝑛
𝐶0 = 1
𝐶1 = 𝑛
𝐶𝑛𝑛 = 1
NÚMEROS COMPLEJOS
𝑖 2 = −1
𝑖 3 = −𝑖
𝑖4 = 1
𝑖 = √−1
√−𝑎 = 𝑖 √𝑎, 𝑎 ≥ 0
(𝑎 + 𝑏𝑖) + (𝑐 + 𝑑𝑖) = 𝑎 + 𝑐 + (𝑏 + 𝑑)𝑖
(𝑎 + 𝑏𝑖) − (𝑐 + 𝑑𝑖) = 𝑎 − 𝑐 + (𝑏 − 𝑑)𝑖
(𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) = 𝑎𝑐 − 𝑏𝑑 + (𝑎𝑑 + 𝑏𝑐)𝑖
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) = 𝑎2 + 𝑏 2
̅̅̅̅̅̅̅̅̅̅̅
(𝑎 + 𝑏𝑖) = 𝑎 − 𝑏𝑖
|𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2
̅̅̅̅̅̅̅̅̅̅̅
(𝑎 + 𝑏𝑖)(𝑎 + 𝑏𝑖) = |𝑎 + 𝑏𝑖|2
1
𝑎 − 𝑏𝑖
𝑎 − 𝑏𝑖
=
= 2
𝑎 + 𝑏𝑖 (𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) 𝑎 + 𝑏 2
VALOR ABSOLUTO
𝑎; 𝑠𝑖 𝑎 ≥ 0
|𝑎| = {
|𝑎| = |−𝑎|
−𝑎; 𝑠𝑖 𝑎 < 0
|𝑎| ≥ 0
|𝑎𝑏| = |𝑎||𝑏|
|𝑎|
𝑎
|𝑎 + 𝑏| ≤ |𝑎| + |𝑏|
| |=
|𝑏|
𝑏
PROPIEDADES DE LOS LOGARITMOS
𝑆𝑖 𝑙𝑜𝑔𝑏 𝑎 = 𝑥 → 𝑎 = 𝑏 𝑥 ; 𝑎 > 0; 𝑏 > 0; 𝑏 ≠ 1
𝑙𝑜𝑔10 𝑎 = 𝑙𝑜𝑔𝑎
𝑙𝑜𝑔𝑒 𝑎 = 𝑙𝑛𝑎
𝑙𝑜𝑔𝑏 𝑏 = 1
𝑙𝑜𝑔𝑏 1 = 0
𝑙𝑜𝑔𝑏 (𝑥 𝑟 ) = 𝑟𝑙𝑜𝑔𝑏 𝑥
𝑙𝑜𝑔𝑏 𝑏 𝑥 = 𝑥
𝑏 𝑙𝑜𝑔𝑏 𝑥 = 𝑥
𝑙𝑜𝑔𝑏 𝑥
𝑙𝑜𝑔𝑎 𝑥 =
𝑙𝑜𝑔𝑏 𝑎
𝑙𝑜𝑔𝑎 𝑏 ⋅ 𝑙𝑜𝑔𝑏 𝑐 ⋅ 𝑙𝑜𝑔𝑐 𝑑 = 𝑙𝑜𝑔𝑎 𝑑
𝑙𝑜𝑔𝑏 (𝑥𝑦) = 𝑙𝑜𝑔𝑏 𝑥 + 𝑙𝑜𝑔𝑏 𝑦
𝑙𝑜𝑔𝑏 𝑥
= 𝑙𝑜𝑔𝑏 𝑎
𝑙𝑜𝑔𝑎 𝑥
𝑥
𝑙𝑜𝑔𝑏 ( ) = 𝑙𝑜𝑔𝑏 𝑥 − 𝑙𝑜𝑔𝑏 𝑦
𝑦
1
𝑐𝑜𝑙𝑜𝑔𝑏 𝑥 = 𝑙𝑜𝑔𝑏 ( ) = 𝑙𝑜𝑔𝑏 (1) − 𝑙𝑜𝑔𝑏 𝑥 = −𝑙𝑜𝑔𝑏 𝑥
𝑥
Versión 1.00
Fórmulas: Jorge.
Diseño: Pedro.
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