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  • Publicado : 19 de agosto de 2012
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1. (2 + x)² = 2² + 2(2)(x) + x²
= 4 + 4x + x²

2. (3a – 5b)² = (3a)² - 2(3a)(5b) + (5b)²
= 9a² - 30ab + 25b²

5. (x + y)² = x² + 2 (x)(y) + y²
= x² + 2xy + y²

6. (p - q)² = p² - 2pq + q²

7. (2p + q)² = (2p)² + 2(2p)(q) + q²
= 4p² + 4pq + q²

8. (3a + b)² = (3a)² + 2(3a)(b) + b²
= 9a² + 6ab + b²

9. (2a - 3b)² = (2a)² - 2(2a)(3b) + (3b)²
= 4a² - 12ab + 9b²

10. (x + 1)² = x²+ 2(x)(1) + (1)²
= x² + 2x + 1

11. (a - 6)² = (a)² - 2(a)(6) + (6)²
= a² - 12a + 36

12. (x + 9)² = (x)² + 2(x)(9) + (9)²
= x² + 18x + 81

13. (3p - 1)² = (3p)² - 2(3p)(1) + (1)²
= 9p² - 6p + 1

14. (x + 5)² = (x)² + 3(x)(5) + (5)²
= x² + 15x + 25

15. (6x – 5y)² = (6x)² - 2(6x)(5y) + (5y)²
= 36x² - 60xy + 25y²

16.- (2m – 1)² = (2m)² - 2(2m)(1) + (1)²
= 4m² - 4m + 1

17.(4pq - 3q)² = (4pq)² - 2(4pq)(3q) + (3q)²
= 16 p² q² – 24pq² + 9q²

18. (9x2 – 7y2)² = (9x²)² - 2(9x²)(7y²) + (7y²)²
= 81x4 - 126 x²y² + 49y4

(8a2b + 7ab6y²)² = (8a²b)² + 2(8a²b)(7ab6y²) + (7ab6y²)²
= 64a4b² + 112a3b7y² + 49a²b12y4

20. (15x²y - 3xy²z6)² =
= (15x²y)² - 2(15x²y)(3xy²z6) + (3x²z6)²
= 225x4y² - 90x²y3z6 + 9x4z12

21. (2a - 3b)² + (3a - 5b)² =
= (4a²-2(2a)(3b)+(9b²)[(3a)²-2(3a)(5b)+ (5b)²]
= 16a² - 12ab + 6b² + 9a² - 30ab + 25b²
= 25a² - 42ab + 31b²

22. (11x - 5y)² - (13x + 3y)² + (x - 2y)²=
= (121x² - 2(11x)(5y) + 25y²) - (169x²+ 2 (13x)(3y) + 9y²) + (x² - 2(x)(2y) + 4y²)
= 121x² - 110xy + 25y² - 169x² - 78xy - 9y² + x² - 4xy + 4y²
= - 47x² - 192xy - 20y²

23. [a/2 + 2b]² + [2a – b/2]² =[(a²/4+2(a/2(2b)+4b²]+[4a²-2(2a)(b/2+b²/4]
= a²/4 + 4ab/² +4b² + 4a² - 4ab/2 + b² /4
= a²/4 + 8ab + 4b² + b²/4
= a² + 16a² /4 + 16b² + b²/4
= 17a²/4 + 17b²/4

24. (3a - b/5)² = 9a² - 2(3a)(b/5) + b²/25
= 9a²- 6ab/5 + b²/25

(2x²/3 - 3yz/5 )² =
= 4x4 /9 - 2(2x²/3)(3yz/5) + (9x²y²/25)
= 4x4 /9 - 12x²yz/15 + 9x²y²/25
= 4x4 /9 - 4x²yz/5 + 9x²y²/25

(0,1a² - 0,2abc)² =
=(1a²/10 - 2abc/10)²
=(1a²/10)² - 2(1a²/10)(2abc/10) + (2abc/10)²
=a4/100 - 4a³bc/100 + 4a²b²c²/100
= 0,01a4 - 0,04a²bc + 0,04a²b²c²

27. (1,5xy² + 2,5xy)² =
= (2.25 x²y4 + 2 (1,5xy²)(2,5x²y) + (6,25x4y²)
= 2.25 x²y4 + 7,5x³y³ + 6,25x4y²

(3a²b³/4 - 3ab6/5 )² =
= 9a4b6/16 -2(3a²b³/4)(3ab6/5)+ (9a²b¹²/25)
= 9a4b6/16 - 18a³b9/20+ 9a²b¹²/25
= 9a4b6/16 - 9a3b9/10 + 9a2b12/25

5.- (4x² – 7xy)² =
6.- (m – 1)² =
7.- (8a + 2ab)² =
8.- (5x + y)² =
9.- (9a –7b)² =
10.- (5ab² + 6)² =
11.- (1 + ab)² =
12.- (5x³y² – x)² =
13.- (5x³y² – 3x)² =
14.- (7x + 7y)² =
15.- (5/6a + 2b)² =

aprender

1) Monomio por monomio a·b = a·b
2) Monomio por polinomio a(c + d) = ac + ad
3) Polinomio por polinomio (a + b)(c + d) = ac + bc + ad + bd
4) Binomio cuadrado (a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
5) Suma por diferencia (a + b)(a – b) = a2 –b2

ejemplos

a) (–4x3y)( –2xy2) = (–4)( –2)( x3x )( yy2 ) = 8x4y3
b) (ab)(4a2b2)( –5a3b4) = 4(–5)( aa2a3 )( bb2b4 ) = –20a6b7




2) Monomio por polinomio a(c + d) = ac + ad
a) 3x(5 – x) = 3x(5) – 3x(x) = 15x – 3x2
b) –2(a – b) = –2a + (–2)( –b) = –2a + 2b




3) Polinomio por polinomio (a + b)(c + d) = ac + bc + ad + bd

Cuadrado de minomio


Polinomio porpolinomio

1) Monomio por monomio | a·b = a·b |
a) (–4x3y)( –2xy2) = (–4)( –2)( x3x )( yy2 ) = 8x4y3 b) (ab)(4a2b2)( –5a3b4) = 4(–5)( aa2a3 )( bb2b4 ) = –20a6b7 |
  
 
2) Monomio por polinomio | a(c + d) = ac + ad |
a) 3x(5 – x) = 3x(5) – 3x(x) = 15x – 3x2 b) –2(a – b) = –2a + (–2)( –b) = –2a + 2b |
1) Factor común monomio  | ac + ad = a(c + d) |
2) Trinomio cuadrado perfecto | a2 + 2ab+ b2 = (a + b)2   a2 – 2ab + b2 = (a – b)2 |
3) Forma an  bn | a2 – b2 = (a + b)(a – b)   a2 + b2 = Irreductible en IR |
4) Trinomio cuadrado perfecto | x2 + (a + b)x + ab = (x + a)(x + b) |
  |

Ejemplos:
 
1) Factor común monomio  | ac + ad = a(c + d) |
 
Factorizar las siguientes expresiones:
a)  6x – 3y = 2(3)x – (3)y = 3(2x – y)
b)  –4xy + 8x = –(4x)y + 2(4x) = 4x(–y + 2)...
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