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### U 5

2. 2. 5Soluciones a los ejercicios y problemas Pág. 2 c) Valor numérico para: x = –1 8 –2 · (–1) = 2 x = 2 8 –2 · 2 = –4 x = 1 8 –2 · 1 = –1 2 2 d) Valor numérico para: x = –1 8 –(–1) 2 = –1 x = 2 8 –2 2 = –4 x= 1 8 – 1 =–1 2 2 2 4 () e) Valor numérico para: x = –1 8 1 (–1) 2 = 1 2 2 x = 2 8 1 · 22 = 1 · 4 = 2 2 2 2 x= 1 8 1 · 1 = 1 · 1 = 1 2 2 2 2 4 8() f ) Valor numérico para: x = –1 8 – 1 (–1) = 1 4 4 x=2 8 –1 ·2=– 1 4 2 x= 1 8 –1 · 1 =–1 2 4 2 8 3 Simplifica. 2 2 a) 2x 6 – 3x 6 – x 6 b) 3x 2 – x + 5x 2 3 1 1 2 c) x – 3x + x d) 2 x 2 – x + x2 2 4 5 10 5 1 e) –2x 3 + x 3 – 3x 3 f ) – x 2 + x 2 + 2x 2 2 2 a) 2x 6 – 3x 6 – x 6 = (2 – 3 – 1)x 6 = –2x 6 3 (3 ) ( ) b) 3x 2 – 2 x 2 + 5x 2 = 3 – 2 + 5 x 2 = 8 – 2 x 2 = 22 x 2 3 3 c) 1 x – 3 x + x = ( 1 – 3 + 1) x = ( 2 – 3 + 4 ) x = 3 x 2 4 2 4 4 4 4 4 x +x =( – + 1)x = ( )x = 13 x 2 2 1 2 22 1 4 2 1 10 2 2 d) x – – + 5 10 5 10 10 10 10 10 e) –2x 3 + x 3 – 3x 3 = (–2 + 1 – 3)x 3 = –4x 3 f) – 5 2 1 2 2 5 1 x + x + 2x 2 = – + 2 2 2 ( 4 ) ( + 2 x 2 = – + 2 x 2 = 0x 2 = 0 2 )
3. 3. 5Soluciones a los ejercicios y problemas Pág. 3 4 Dados los monomios A = –5x 4, B= 20x 4, C = 2x, calcula: a) A + B b) A – B c) 3A + 2B d) A 3 e) C2 f) A2 + C8 g) A · B h)A · C i) B · C j) B : A k) A : B l) (B : C ) · A A = –5x 4 B = 20x 4 C = 2x a) A + B = –5x 4 + 20x 4 = 15x 4 b) A – B = –5x 4 – 20x 4 = –25x 4 c) 3A + 2B = 3 · (–5x 4) + 2 · (20x 4) = –15x 4 + 40x 4 = 25x 4 d) A 3 = (–5x 4) 3 = –125x 12 e) C 2 = (2x) 2 = 4x 2 f ) A 2 + C 8 = (–5x 4) 2 + (2x) 8 = 25x 8 + 256x 8 = 281x 8 g) A · B = (–5x 4) · (20x 4) = –100x 8 h) A · C = (–5x 4) · (2x) = –10x 5 i) B · C = (20x 4) · (2x) = 40x 5 j) B : A = (20x 4) : (–5x 4) = –4 k) A : B = (–5x 4) : (20x 4) = – 5 = – 1 20 4 4 l) (B : C ) · A = 20x · (–5x 4) = (10x 3) · (–5x 4) = –50x 7 2x 5 Efectúa las siguientes operaciones y di cuál es el grado del monomio resul- tante: a) 2x · (–3x 2) · (–x) b) 3 x 3 · (–2x 2) · 2x 4 c) 2x 3 · (–x 2) · 5x ( ) d) x · – 1 x · 3 x 2 5 e) – 1 x · 3x 2 · (–x) f ) 2 x 2 · 3 x · 10 x 2 3 5 4 3 a) 2x · (–3x 2) · (–x) = 6x 4 8 Grado 4 b) 3 x 3 · (–2x 2) · 2x = 3 · (–4)x 6 = –3x 6 8 Grado 6 4 4 c) 2x 3 · (–x 2) · 5x = –10x 6 8 Grado 6 ( ) d) x · – 1 x · 3 x = – 3 x 3 8 Grado 3 2 5 10 e) – 1 x · 3x 2 · (–x) = x 4 8 Grado 4 3 f ) 2 x 2 · 3 x · 10 x 2 = 2 · 3 · 10 · x 5 = x 5 8 Grado 5 5 4 3 5 4 3
5. 5. 5Soluciones a los ejercicios y problemas Pág. 5 8 Dados los polinomios P = 2x 4 – 5x 3 + 3x – 1 y Q = 6x 3 + 2x 2 – 7, calcula P + Q y P – Q. P + Q = (2x 4 – 5x 3 + 3x – 1) + (6x 3 + 2x 2 – 7) = 2x 4 + x 3 + 2x 2 + 3x – 8 P – Q = (2x 4 – 5x 3 + 3x – 1) – (6x 3 + 2x 2 – 7) = 2x 4 – 5x 3 + 3x – 1 – 6x 3 – 2x 2 + 7 = = 2x 4 – 11x 3 – 2x 2 + 3x + 6 9 Sean los polinomios: M = 3x 2 – 5x – 3 N = 1 x2 + 3 x + 1 K = x2 – 1 x + 2 2 4 3 3 Calcula: a) 2M + 3K b) M – 4N c) 4N – 3K ( ) a) 2M + 3K = 2(3x 2 – 5x – 3) + 3 x 2 – 1 x + 2 = 6x 2 – 10x – 6 + 3x 2 – x + 2 = 3 3 = 9x 2 – 11x – 4 ( ) b) M – 4N = (3x 2 – 5x – 3) – 4 1 x 2 + 3 x + 1 = 3x 2 – 5x – 3 – 2x 2 – 3x – 4 = 2 4 = x 2 – 8x – 7 2 ( 4 ) ( 3 3 ) c) 4N – 3K = 4 1 x 2 + 3 x + 1 – 3 · x 2 – 1 x + 2 = 2x 2 + 3x + 4 – 3x 2 + x – 2 = = –x 2 + 4x + 2 10 Efectúa. a) 3x (2x 2 – 5x + 1) b) 7x 3(2x 3 + 3x 2 – 2) c) –5x (x 4 – 3x 2 + 5x) d) –x 2(x 3 + 4x 2 – 6x + 3) a) 3x (2x 2 – 5x + 1) = 6x 3 – 15x 2 + 3x b) 7x 3(2x 3 + 3x 2 – 2) = 14x 6 + 21x 5 – 14x 3 c) –5x (x 4 – 3x 2 + 5x) = –5x 5 + 15x 3 – 25x 2 d) –x 2(x 3 + 4x 2 – 6x + 3) = –x 5 – 4x 4 + 6x 3 – 3x 2 11 Opera y simplifica: a) (5x – 2) (3 – 2x) b) x (x – 3) (2x – 1) c) (3 + 7x)(5 + 2x) d) (x + 1)(3x + 2)(x – 2) a) (5x – 2)(3 – 2x) = 15x – 10x 2 – 6 + 4x = –10x 2 + 19x – 6 b) x (x – 3)(2x – 1) = (x 2 – 3x)(2x – 1) = 2x 3 – x 2 – 6x 2 + 3x = 2x 3 – 7x 2 + 3x c) (3 + 7x)(5 + 2x) = 15 + 6x + 35x + 14x 2 = 14x 2 + 41x +15 d) (x + 1)(3x + 2)(x – 2) = (3x 2 + 2x + 3x + 2)(x – 2) = (3x 2 + 5x + 2)(x – 2) = = 3x 3 + 5x 2 + 2x – 6x 2 – 10x – 4 = 3x 3 – x 2 – 8x – 4
6. 6. 5Soluciones a los ejercicios y problemas Pág. 6 12 Opera y simplifica: a) (3x 3 + 1)(2x 2 – 3x + 5) b) (x 2 – 5x) (x 3 + 2x) c) (x 3 – 2x + 3) (x 2 + 4x – 1) d) (3x 2 – 2x + 2) (x 3 + 3x – 2) a) (3x 3 + 1)(2x 2 – 3x + 5) = 6x 5 – 9x 4 + 15x 3 + 2x 2 – 3x + 5 b) (x 2 – 5x) · (x 3 + 2x) = x 5 + 2x 3 – 5x 4 – 10x 2 c) (x 3 – 2x + 3) · (x 2 + 4x – 1) = = x 5 + 4x 4 – x 3 – 2x 3 – 8x 2 + 2x + 3x 2 + 12x – 3 = = x 5 + 4x 4 – 3x 3 – 5x 2 + 14x – 3 d) (3x 2 – 2x + 2) · (x 3 + 3x – 2) = = 3x 5 + 9x 3 – 6x 2 – 2x 4 – 6x 2 + 4x + 2x 3 + 6x – 4 = = 3x 5 – 2x 4 + 11x 3 – 12x 2 + 10x – 4 PÁGINA 92 13 Calcula el cociente y el resto en cada una de estas divisiones: a) (x 5 + 7x 3 – 5x + 1) : x b) (x 3 – 5x 2 + x) : (x – 2) c) (x 3 – 5x 2 + x) : (x + 3) a) x 5 + 7x 3 – 5x + 1 |x –x 5 x 4 + 7x 2 – 5 7x 3 –7x 3 –5x 5x +1 Cociente = x 4 + 7x 2 – 5 Resto = 1 b) x 3 – 5x 2 + x |x–2 –x 3 + 2x 2 x 2 – 3x – 5 –3x 2 + x 3x 2 – 6x –5x 5x – 10 –10 Cociente = x 2 – 3x – 5 Resto = –10
7. 7. 5Soluciones a los ejercicios y problemas Pág. 7 c) x3 – 5x 2 + x |x +3 –x 3 – 3x 2 x 2 – 8x + 25 –8x 2 + x 8x 2 + 24x 25x –25x – 75 –75 Cociente = x 2 – 8x + 25 Resto = –75 14 Halla el cociente y el resto en cada una de estas divisiones: a) (3x 2 – 7x + 5) : (3x + 1) b) (4x 3 – x) : (2x + 3) c) (5x 3 – 3x 2 + 8x) : (5x + 2) a) 3x 2 – 7x + 5 | 3x + 1 –3x 2 – x x– 8 3 –8x + 5 8x + 8 3 23 3 Cociente = x – 8 Resto = 23 3 3 b) 4x 3 – x | 2x + 3 –4x 3 – 6x 2 2x 2 – 3x + 4 –6x 2 – x 6x 2 + 9x 8x –8x – 12 –12 Cociente = 2x 2 – 3x + 4 Resto = –12 c) 5x 3 – 3x 2 + 8x | 5x + 2 –5x 3 – 2x 2 x2 – x + 2 –5x 2 + 8x 5x 2 + 2x 10x –10x – 4 –4 Cociente = x 2–x +2 Resto = –4
8. 8. 5Soluciones a los ejercicios y problemas Pág. 8 Factorización de polinomios 15 Saca factor común en cada caso: a) 9x 2 + 6x – 3 b) 2x 3 – 6x 2 + 4x c) 10x 3 – 5x 2 d) x 4 – x 3 + x 2 – x a) 9x 2 + 6x – 3 = 3(3x 2 + 2x – 1) b) 2x 3 – 6x 2 + 4x = 2x(x 2 – 3x + 2) c) 10x 3 – 5x 2 = 5x 2 (2x – 1) d) x 4 – x 3 + x 2 – x = x(x 3 – x 2 + x – 1) 16 Saca factor común en cada polinomio: a) 410x 5 – 620x 3 + 130x b) 72x 4 – 64x 3 c) 5x – 100x 3 d) 30x 6 – 75x 4 – 45x 2 a) 410x 5 – 620x 3 + 130x = 10x(41x 4 – 62x 2 + 13) b) 72x 4 – 64x 3 = 8x 3(9x – 8) c) 5x – 100x 3 = (1 – 20x 2) d) 30x 6 – 75x 4 – 45x 2 = 15x 2(2x 4 – 5x 2 – 3) 17 Expresa los polinomios siguientes como cuadrado de un binomio: a) x 2 n + 12x + 36 = (x + I ) 2 n b) 4x 2 – 20x + 25 = ( I – 5) 2 c) 49 + 14x + x 2 d) x 2 – x + 1 4 a) x 2 + 12x + 36 = (x + 6) 2 b) 4x 2 – 20x + 25 = (2x – 5) 2 c) 49 + 14x + x 2 = (7 + x) 2 4 ( ) d) x 2 – x + 1 = x – 1 2 2 18 Expresa como producto de dos binomios los siguientes polinomios: a) x 2 n n – 16 = (x + I ) (x – I ) b) x 2 – 1 c) 9 – x 2 d) 4x 2 – 1 e) 4x 2 – 9 a) x 2 – 16 = (x + 4)(x – 4) b) x 2 – 1 = (x + 1)(x – 1) c) 9 – x 2 = (3 + x)(3 – x) d) 4x 2 – 1 = (2x – 1)(2x + 1) e) 4x 2 – 9 = (2x – 3)(2x + 3)
9. 9. 5Soluciones a los ejercicios y problemas Pág. 9 19 Expresa como un cuadrado o como producto de dos binomios cada uno de los siguientes polinomios: a) 25x 2 + 40x + 16 b) 64x 2 – 160x + 100 c) 4x 2 – 25 d) x 4 – 1 a) 25x 2 + 40x + 16 = (5x) 2 + 2 · 5x · 4 + 4 2 = (5x + 4) 2 b) 64x 2 – 160x + 100 = (8x) 2 – 2 · 8x · 10 + 10 2 = (8x – 10) 2 c) 4x 2 – 25 = (2x) 2 – 5 2 = (2x + 5)(2x – 5) d) x 4 – 1 = (x 2 – 1)(x 2 + 1) = (x + 1)(x – 1) (x 2 + 1) En realidad, se puede poner como producto de tres binomios. 20 Saca factor común y utiliza los productos notables para factorizar los si- guientes polinomios: a) x 3 – 6x 2 + 9x b) x 3 – x c) 4x 4 – 81x 2 d) x 3 + 2x 2 + x e) 3x 3 – 27x f ) 3x 2 + 30x + 75 a) x 3 – 6x 2 + 9x = x(x 2 – 6x + 9) = x(x – 3) 2 b) x 3 – x = x(x 2 – 1) = x(x – 1)(x + 1) c) 4x 4 – 81x 2 = x 2 (4x 2 – 81) = x 2 (2x + 9)(2x – 9) d) x 3 + 2x 2 + x = x(x 2 + 2x + 1) = x(x + 1) 2 e) 3x 3 – 27x = 3x(x 2 – 9) = 3x(x + 3)(x – 3) f ) 3x 2 + 30x + 75 = 3(x 2 + 10x + 25) = 3(x + 5) 2 Expresiones de primer grado 21 Simplifica. a) 6(x + 3) – 2(x – 5) b) 3(2x + 1) + 7(x – 3) – 4x c) 5(3 – 2x) – (x + 7) – 8 d) 4(1 – x) + 6x – 10 – 3(x – 5) e) 2x – 3 + 3(x – 1) – 2(3 – x) + 5 f ) 2(x + 3) – (x + 1) – 1 + 3(5x – 4) a) 6(x + 3) – 2(x – 5) = 6x + 18 – 2x + 10 = 4x + 28 b) 3(2x + 1) + 7(x – 3) – 4x = 6x + 3 + 7x – 21 – 4x = 9x – 18 c) 5(3 – 2x) – (x + 7) – 8 = 15 – 10x – x – 7 – 8 = –11x d) 4(1 – x) + 6x – 10 – 3(x – 5) = 4 – 4x + 6x – 10 – 3x + 15 = –x + 9 e) 2x – 3 + 3(x – 1) – 2(3 – x) + 5 = 2x – 3 + 3x – 3 – 6 + 2x + 5 = 7x – 7 f ) 2(x + 3) – (x + 1) – 1 + 3(5x – 4) = 2x + 6 – x – 1 – 1 + 15x – 12 = 16x – 8
10. 10. 5Soluciones a los ejercicios y problemas Pág. 10 22 Multiplica por el número indicado y simplifica. a) 1 – 2x – 1 + x + 4 por 18 9 6 b) 3x + 2 – 4x – 1 + 5x – 2 – x + 1 por 40 5 10 8 4 c) x – 3 – 5x + 1 – 1 – 9x por 6 2 3 6 d) x + 1 + x – 3 – 2x + 6 – x – 8 por 10 2 5 5 e) 1 + 12x + x – 4 – 3(x + 1) – (1 – x) por 8 4 2 8 f ) 3x – 2 – 4x + 1 + 2 + 2(x – 3) por 60 6 10 15 4 ( ) a) 18 1 – 2x – 1 + x + 4 = 2(1 – 2x) – 18 + 3(x + 4) = 2 – 4x – 18 + 3x + 12 = 9 6 = –x – 4 ( b) 40 3x + 2 – 4x – 1 + 5x – 2 – x + 1 = 5 10 8 4 ) = 8(3x + 2) – 4(4x – 1) + 5(5x – 2) – 10(x + 1) = = 24x + 16 – 16x + 4 + 25x – 10 – 10x – 10 = 23x ( 2 3 6 ) c) 6 x – 3 – 5x + 1 – 1 – 9x = 3(x – 3) – 2(5x + 1) – (1 –9x) = = 3x – 9 – 10x – 2 – 1 + 9x = 2x – 12 (2 5 5 ) d) 10 x + 1 + x – 3 – 2x + 6 – x – 8 = 5(x + 1) + 2(x – 3) – 20x + 6 – 2(x – 8) = = 5x + 5 + 2x – 6 – 20x + 60 – 2x + 16 = = –15x + 75 ( e) 8 1 + 12x + x – 4 – 3(x + 1) – (1 – x) = 4 2 8 ) = 2(1 + 12x) + 4(x – 4) – 3(x + 1) + (1 – x) = = 2 + 24x + 4x – 16 – 3x – 3 + 1 – x = 24x – 16 ( f ) 60 3x – 2 – 4x + 1 + 2 + 2(x – 3) = 6 10 15 4 ) = 10(3x – 2) – 6(4x + 1) + 4 · 2 + 15 · 2(x – 3) = = 30x – 20 – 24x – 6 + 8 + 30x – 90 = 36x – 108
11. 11. 5Soluciones a los ejercicios y problemas Pág. 11 Expresiones de segundo grado 23 Simplifica las siguientes expresiones: a) (x – 3)(x + 3) + (x – 4)(x + 4) – 25 b) (x + 1)(x – 3) + (x – 2)(x – 3) – (x 2 – 3x – 1) c) 2x (x + 3) – 2(3x + 5) + x d) (x + 1)2 – 3x – 3 e) (2x + 1)2 – 1 – (x – 1)(x + 1) f ) x (x – 3) + (x + 4)(x – 4) – (2 – 3x) a) (x – 3)(x + 3) + (x – 4)(x + 4) – 25 = x 2 – 9 + x 2 – 16 – 25 = 2x 2 – 50 b) (x + 1)(x – 3) + (x – 2)(x – 3) – (x 2 – 3x – 1) = = x 2 – 3x + x – 3 + x 2 – 3x – 2x + 6 – x 2 + 3x + 1 = x 2 – 4x + 4 c) 2x (x + 3) – 2(3x + 5) + x = 2x 2 + 6x – 6x – 10 + x = 2x 2 + x – 10 d) (x + 1)2 – 3x – 3 = x 2 + 2x + 1 – 3x – 3 = x 2 – x – 2 e) (2x + 1)2 – 1 – (x – 1)(x + 1) = 4x 2 + 4x + 1 – 1 – (x 2 – 1) = = 4x 2 + 4x – x 2 + 1 = 3x 2 + 4x + 1 f ) x (x – 3) + (x + 4)(x – 4) – (2 – 3x) = x 2 – 3x + x 2 – 16 – 2 + 3x = 2x 2 – 18 PÁGINA 93 24 Multiplica por el número indicado y simplifica. 2 a) (3x + 1)(3x – 1) + (x – 2) – 1 + 2x por 2 2 2 2 b) x + 2 – x + 1 – x + 5 por 12 3 4 12 2 c) (2x – 1)(2x + 1) – 3x – 2 – x por 6 3 6 3 d) (x + 1)(x – 3) + x – x por 4 2 4 e) x + 3x + 1 – x – 2 – x 2 + 2 por 6 2 3 f ) x (x – 1) – x (x + 1) + 3x + 4 por 12 3 4 12 ( 2 ) a) 2 (3x + 1)(3x – 1) + (x – 2) – 1 + 2x = 2(3x + 1)(3x – 1) + (x – 2)2 – 2 + 4x = 2 = 2(9x 2 – 1) + x 2 – 4x + 4 – 2 + 4x = = 18x 2 – 2 + x 2 + 2 = 19x 2
12. 12. 5Soluciones a los ejercicios y problemas Pág. 12 b) 12 ( x2 +2– 3 x2 ) + 1 – x + 5 = 4(x 2 + 2) – 3(x 2 + 1) – (x + 5) = 4 12 = 4x 2 + 8 – 3x 2 – 3 – x – 5 = x 2 – x ( 2 ) c) 6 (2x – 1)(2x + 1) – 3x – 2 – x = 2(2x – 1)(2x + 1) – (3x – 2) – 2x 2 = 3 6 3 = 2(4x 2 – 1) – 3x + 2 – 2x 2 = = 8x 2 – 2 – 3x + 2 – 2x 2 = 6x 2 – 3x ( ) d) 4 (x + 1)(x – 3) + x – x = 2(x + 1)(x – 3) + 4x – x = (2x + 2)(x – 3) + 3x = 2 4 = 2x 2 – 6x + 2x – 6 + 3x = 2x 2 – x – 6 ( ) e) 6 x + 3x + 1 – x – 2 – x 2 + 2 = 6x + 3(3x + 1) – 2(x – 2) – 6x 2 + 12 = 2 3 = 6x + 9x + 3 – 2x + 4 – 6x 2 + 12 = = –6x 2 + 13x + 19 ( 3 4 12 ) f ) 12 x (x – 1) – x (x + 1) + 3x + 4 = 4x(x – 1) – 3x(x + 1) + 3x + 4 = = 4x 2 – 4x – 3x 2 – 3x + 3x + 4 = = x 2 – 4x + 4 Expresiones no polinómicas 25 Desarrolla A 2 – B 2 y simplifica en cada uno de los siguientes casos: a) A = √x , B = x – 2 b) A = √25 – x 2 , B = x – 1 c) A = √169 – x 2 , B = x – 17 d) A = √5x + 10, B = 8 – x e) A = √2x 2 + 7, B = √5 – 4x f ) A = √x + 2 , B = x – 4 a) A = √x , B = x – 2 (√x )2 – (x – 2)2 = x – (x 2 – 4x + 4) = x – x 2 + 4x – 4 = –x 2 + 5x – 4 b) A = √25 – x 2 , B = x – 1 (√25 – x 2 )2 – (x – 1)2 = 25 – x 2 – (x 2 – 2x + 1) = 25 – x 2 – x 2 + 2x – 1 = = –2x 2 + 2x + 24 c) A = √169 – x 2 , B = x – 17 (√169 – x 2 )2 – (x – 17)2 = 169 – x 2 – (x 2 – 34x + 289) = = 169 – x 2 – x 2 + 34x – 289 = –2x 2 + 34x – 120
13. 13. 5Soluciones a los ejercicios y problemas Pág. 13 d) A = √5x + 10 , B = 8 – x (√5x + 10 )2 – (8 – x)2 = 5x + 10 – (64 – 16x + x 2) = 5x + 10 – 64 + 16x – x 2 = = –x 2 + 21x – 54 e) A = √2x 2 + 7 , B = √5 – 4x (√2x 2 + 7 )2 – (√5 – 4x )2 = 2x 2 + 7 – (5 – 4x) = 2x 2 + 7 – 5 + 4x = 2x 2 + 4x + 2 f ) A = √x + 2 , B = x – 4 (√x + 2 )2 – (x – 4)2 = x + 2 – (x 2 – 8x + 16) = x + 2 – x 2 + 8x – 16 = = –x 2 + 9x – 14 26 Multiplica por la expresión indicada y simplifica. a) 2 – 1 – 3x por 2x x 2x 2 b) 800 – 50 – 600 por x (x + 4) x x+4 c) 12 – 2 – 3 – 2x por 3x 2 x 3x d) x – 1 – 2x – 4 por 2(x + 4) 2 x+4 e) 100 + 5 – 90 por x (x – 4) x x–4 f ) 250 – 5 – 3(4x – 1) por x + 1 x+1 g) 1 + 22 – 5 por 9x 2 x x 9 h) 2 – x + 4 – 1 por 2(2 + x) 2 2+x ( ) a) 2x 2 – 1 – 3x = 4 – 1 – 3x 2 = –3x 2 + 3 x 2x 2 ( ) b) x (x + 4) 800 – 50 – 600 = 800(x + 4) – 50x(x + 4) – 600x = x x+4 = 800x + 3 200 – 50x 2 – 200x – 600x = –50x 2 + 3 200 ( x 3x x ) c) 3x 2 12 – 2 – 3 – 2 = 3 – 6x 2 – 3 + x = –6x 2 + x ( ) d) 2(x + 4) x – 1 – 2x – 4 = x(x + 4) – 2(x + 4) – 2(2x – 4) = 2 x+4 = x 2 + 4x – 2x – 8 – 4x + 8 = x 2 – 2x ( ) e) x (x – 4) 100 + 5 – 90 = 100(x – 4) + 5x(x – 4) – 90x = x x–4 = 100x – 400 + 5x 2 – 20x – 90x = 5x 2 – 10x – 400
14. 14. 5Soluciones a los ejercicios y problemas Pág. 14 ( ) f ) (x + 1) 250 – 5 – 3(4x – 1) = 250 – 5(x + 1) – 3(x + 1)(4x – 1) = x+1 = 250 – 5x – 5 – (3x + 3)(4x – 1) = 250 – 5x – 5 – (12x 2 – 3x + 12x – 3) = = 250 – 5x – 5 – 12x 2 + 3x – 12x + 3 = –12x 2 – 14x + 248 ( ) g) 9x 2 1 + 22 – 5 = 9x + 18 – 5x 2 = –5x 2 + 9x + 18 x x 9 ( ) h) 2(2 + x) 2 – x + 4 – 1 = (2 + x)(2 – x) + 4 · 2 – 2(2 + x) = 2 2+x = 4 – x 2 + 8 – 4 – 2x = –x 2 – 2x + 8 P I E N S A Y R E S U E LV E Monomios, polinomios, factorización 27 Al multiplicar P (x) por 3x 2 hemos obtenido –15x 4. ¿Cuánto vale P (x)? –15x 4 Si P(x) · 3x 2 = –15x 4 8 P(x) = = –5x 2 3x 2 28 Al dividir M (x) entre 2x 3 hemos obtenido 5x 2. ¿Cuánto vale M (x)? Si M(x): 2x 3 = 5x 2 8 M(x) = 5x 2 · 2x 3 = 10x 5 29 Calcula un polinomio P (x) tal que: A (x) + P (x) = 2x 4 + x 3 + x 2 – 2x – 1 siendo: A (x) = x 3 + 3x 2 – 5x – 1 P(x) = (2x 4 + x 3 + x 2 – 2x – 1) – A(x) = = (2x 4 + x 3 + x 2 – 2x – 1) – (x 3 + 3x 2 – 5x – 1) = = 2x 4 + x 3 + x 2 – 2x – 1 – x 3 – 3x 2 + 5x + 1 = 2x 4 – 2x 2 + 3x 30 Calcula un polinomio P (x) tal que: 3A (x) – P (x) = –5x 3 + 3x 2 – 2x + 5 siendo: A (x) = x 2 – 2x + 1 P(x) = 3A(x) – (–5x 3 + 3x 2 – 2x + 5) = 3(x 2 – 2x + 1) + 5x 3 – 3x 2 + 2x – 5 = = 3x 2 – 6x + 3 + 5x 3 – 3x 2 + 2x – 5 = 5x 3 – 4x – 2 31 Calcula un polinomio P(x) tal que: A(x) – 2B(x) + P(x) = x 4 + x 3 + x 2 + x + 1 siendo: A (x) = 2x 4 – 3x 2 – 4x + 5 B (x) = x 3 – 5x 2 – 5x + 9 Despejamos P(x) de la expresión dada; así: P(x) = x 4 + x 3 + x 2 + x + 1 – A(x) + 2B(x) P(x) = x 4 + x 3 + x 2 + x + 1 – (2x 4 – 3x 2 – 4x + 5) + 2(x 3 – 5x 2 – 5x + 9) P(x) = x 4 + x 3 + x 2 + x + 1 – 2x 4 + 3x 2 + 4x – 5 + 2x 3 – 10x 2 – 10x + 18 P(x) = –x 4 + 3x 3 – 6x 2 – 5x + 14
15. 15. 5Soluciones a los ejercicios y problemas Pág. 15 32 Efectúa las siguientes divisiones y expresa el resultado de la forma P (x) = Q (x) · C (x) + R (x): a) (x 2 – 3x + 2) : (x + 4) b) (x 3 – 2x + 3) : (x 2 – 1) c) (3x 2 – 2x + 7) : (x – 2) d) (x 2 + x – 12) : (x + 3) a) (x 2 – 3x + 2) : (x + 4) x 2 – 3x + 2 |x+4 –x 2 – 4x x–7 C (x) = x – 7 –7x + 2 7x + 28 30 R(x) = 30 Por tanto: x 2 – 3x + 2 = (x + 4)(x – 7) + 30 b) (x 3 – 2x + 3) : (x 2 – 1) x 3 – 2x + 3 | x2 – 1 –x 3 + x x C (x) = x –x + 3 R(x) = –x + 3 Así: x 3 – 2x + 3 = (x 2 – 1)x – x + 3 c) (3x 2 – 2x + 7) : (x – 2) 3x 2 – 2x + 7 |x–2 –3x 2 + 6x 3x + 4 C (x) = 3x + 4 4x + 7 –4x + 8 15 R(x) = 15 Por tanto: 3x 2 – 2x + 7 = (x – 2)(3x + 4) + 15 d) (x 2 + x – 12) : (x + 3) x 2 + x – 12 |x+3 –x 2 – 3x x–2 C (x) = x – 2 –2x – 12 2x + 6 –6 R(x) = –6 Por tanto: x 2 + x – 12 = (x + 3)(x – 2) – 6
16. 16. 5Soluciones a los ejercicios y problemas Pág. 16 33 Las siguientes divisiones son exactas. Efectúalas y expresa el dividendo como producto de dos factores: a) (x 5 + 2x 4 + x + 2) : (x + 2) b) (3x 3 + 7x 2 + 7x + 4) : (3x + 4) c) (x 3 – x 2 + 9x – 9) : (x – 1) d) (2x 3 – 3x 2 + 10x – 15) : (2x – 3) a) (x 5 + 2x 4 + x + 2) : (x + 2) x 5 + 2x 4 + x + 2 |x+2 –x 5 – 2x 4 x4 + 1 x +2 –x – 2 0 Por tanto: x 5 + 2x 4 + x + 2 = (x + 2)(x 4 + 1) b) (3x 3 + 7x 2 + 7x + 4) : (3x + 4) 3x 3 + 7x 2 + 7x + 4 | 3x + 4 –3x 3 – 4x 2 x2 + x + 1 3x 2 + 7x –3x 2 – 4x 3x + 4 –3x – 4 0 Por tanto: 3x 3 + 7x 2 + 7x + 4 = (3x + 4)(x 2 + x + 1) c) (x 3 – x 2 + 9x – 9) : (x – 1) x 3 – x 2 + 9x – 9 |x –1 –x 3 + x 2 x2 + 9 9x – 9 –9x + 9 0 Por tanto: x 3 – x 2 + 9x – 9 = (x – 1)(x 2 + 9) d) (2x 3 – 3x 2 + 10x – 15) : (2x – 3) 2x 3 – 3x 2 + 10x – 15 | 2x – 3 –2x 3 + 3x 2 x2 + 5 10x – 15 –10x + 15 0 Por tanto: 2x 3 – 3x 2 + 10x – 15 = (2x – 3)(x 2 + 5)