Factorice cada uno de los siguientes:
Pregunta 1. a 3 + 8b 3 + 64c 3 – 24abc
Solución:
Lo sabemos
a 3 + b 3 + c 3 – 3abc = (a + b + c) (a 2 + b 2 + c 2 – ab – bc – ca)
= a 3 + 8b 3 + 64c 3 – 24abc
= (a) 3 + (2b) 3 + (4c) 3 – (3 × a × 2b × 4c)
= (a + 2b + 4c) [(a) 2 + (2b) 2 + (4c) 2 -(a × 2b)– (2b × 4c)– (4c × a)] 2
= (a + 2b + 4c) (a 2 + 4b 2 + 16c 2 – 2ab – 8bc – 4ca)
Pregunta 2. x 3 – 8y 3 + 27z 3 + 18xyz
Solución:
Podemos simplificar la ecuación dada como:
x 3 – 8y 3 + 27z 3 + 18xyz
= (x) 3 + (-2y) 3 + (3z) 3 – ( 3 * x * (-2y) * (3 z))
= (x – y + 3z) (x2 + 4y2 + 9z2 + 2xy + 6yz – 3zx)
Pregunta 3. 27x 3 – y 3 – z 3 – 9xyz
Solución:
27x 3 – y 3 – z 3 – 9xyz
= (3x) 3 + (-y) 3 + (-z) 3 – (3 * 3x * (-y) (-z))
= (3x – y – z) [(3x) 2 + (-y) 2 + (-z) 2 – (3x * (-y)) – ((-y) (-z))- (- z × 3x)]
= (3x – y – z) (9x 2 + y 2 + z 2 + 3xy – yz + 3zx)
Pregunta 4. (1/27)x 3 – y 3 + 125z 3 + 5xyz
Solución:
=(1/3x) 3 + -(y) 3 +(5z) 3 – 3*(1/3x) * (-y) * (5z)
=((1/3x) – y +5z)[(1/3x) 2 + (-y) 2 + (5z) 2 -((1/3x)*-y) – (-y * 5z) – ( 5z * (1/3x))
= ((1/3x) – y +5z)[(1/9)x 2 + y 2 +25z 2 + (1/3xy) + 5yz – (5/3xz)]
Pregunta 5. 8x 3 + 27y 3 – 216z 3 + 108xyz
Solución:
8x 3 + 27y 3 – 216z 3 + 108xyz
= (2x) 3 + (3y) 3 + (6z) 3 – 3 × (2x) (3y) (-6z)
= (2x + 3y – 6z) [(2x) 2 + (3y) 2 + (-6z) 2 – (2x * 3y) – (3y * (-6z)) – ((-6z) * 2x)]
= (2x + 3y – 6z) (4x 2 + 9y 2 + 36z 2 – 6xy + 18yz + 12zx)
Pregunta 6. 125 + 8x 3 – 27y 3 + 90xy
Solución:
125 + 8X 3 – 27y 3 + 90xy
= (5) 3 + (2x) 3 + (-3y) 3 – [3 * 5 * 2x * (-3y)]
= (5 + 2x – 3y) [(5) 2 + (2x) 2 + (-3y) 2 – (5 * 2x) – (2x * (-3y)) – ((-3y) * 5)]
= (5 + 2x – 3y) (25 + 4x 2 + 9y 2 – 10x + 6xy + 15y)
Pregunta 7. 8x 3 – 125y 3 + 180xy + 216
Solución:
8x 3 – 125y 3 + 180xy + 216
= (2x) 3 + (-5y) 3 + (6) 3 – 3 * 2x *(-5y) * 6
= (2x – 5y + 6) [(2x) 2 + (-5y) 2 + (6) 2 – 2x *(-5y) – (-5y) * 6 – 6 * 2x]
= (2x – 5y + 6) (4x 2 + 25y 2 + 36 + 10xy + 30y – 12x)
Pregunta 8. Multiplica:
(i) x 2 + y 2 + z 2 – xy + xz + yz por x + y – z
(ii) x 2 + 4y 2 + z 2 + 2xy + xz – 2yz por x- 2y-z
(iii) x 2 + 4y 2 + 2xy – 3x + 6y + 9 por x – 2y + 3
(iv) 9x 2 + 25y 2 + 15xy + 12x – 20y + 16 por 3x – 5y + 4
Solución:
(i) (x 2 + y 2 + z 2 – xy + yz + zx) por (x + y – z)
= x 3 + y 3 – z 3 + 3xyz
(ii) (x 2 + 4y 2 + z 2 + 2xy + xz – 2yz) por (x – 2y – z)
= (x -2y-z) [x 2 + (-2y) 2 + (-z) 2 – (x * (- 2y)) – ((-2y)* (z)) – ((-z) ( X))]
= x 3 + (-2y) 3 + (-z) 3 – 3x * (-2y) * (-z)
= x 3 – 8y 3 – z 3 – 6xyz
(iii) x 2 + 4y 2 + 2xy – 3x + 6y + 9 por (x – 2y + 3)
= (x – 2y + 3) (x 2 + 4y 2 + 9 + 2xy + 6y – 3x)
= (x) 3 + (-2y) 3 + (3) 3 – (3 * x * (-2y) x 3)
= x 3 – 8y 3 + 27 + 18xy
(iv) 9x 2 + 25y 2 + 15xy + 12x – 20y + 16 por (3x – 5y + 4)
= (3x -5y + 4) [(3x) 2 + (-5y) 2 + (4) 2 – 3x * (-5y) +(-5y x 4) + (4 × 3x)]
= (3x) 3 + (-5y) 3 + (4) 3 – 3 * 3x *(-5y) * 4
= 27x 3 – 125y 3 + 64 + 180xy
Pregunta 9. (3x – 2y) 3 + (2y – 4z) 3 + (4z – 3x) 3
Solución:
(3x – 2y) 3 + (2y – 4z) 3 + (4z – 3x) 3
∵ 3x – 2y + 2y – 4z + 4z – 3x = 0
∴ (3x – 2y) 3 + (2y – 4z) 3 + (4z – 3x) 3
= 3(3x – 2y) (2y – 4z) (4z – 3x) {∵ x 3 + y 3 + z 3 = 3xyz si x + y + z = 0}
Pregunta 10. (2x – 3y) 3 + (4z – 2x) 3 + (3y – 4z) 3
Solución:
(2x – 3y) 3 + (4z – 2x) 3 + (3y – 4z) 3
∵ 2x – 3y + 4z – 2x + 3y – 4z = 0
∴ (2x – 3y) 3 + (4z – 2x) 3 + (3y – 4z) 3
= 3(2x – 3y) (4z – 2x) (3y – 4z) {∵ x 3 + y 3 + z 3 = 3xyz si x + y + z = 0}
Pregunta 11. [(x/2)+y +(z/3)] 3 + [(x/3) -(2y/3) +z] 3 + [(-5x/6)-(y/3) -(4z/3)] 3
Solución:
[(x/2)+y +(z/3)] 3 + [(x/3) -(2y/3) +z] 3 + [(-5x/6)-(y/3)-(4z /3)] 3
∵ (x/2) + y +(z/3) +(x/3) -(2y/3) + z – (5x/6) -(y/3) – (4z/3) =0
∴ [(x/2)+y +(z/3)] 3 + [(x/3) -(2y/3) +z] 3 + [(-5x/6)-(y/3)-( 4z/3)] 3
= 3[(x/2)+y +(z/3)] [(x/3) -(2y/3) +z] [(-5x/6)-(y/3)-(4z/3 )] {∵ a 3 + b 3 + c 3 = 3abc si a + b + c = 0}
Pregunta 12. (a – 3b) 3 + (3b – c) 3 + (c – a) 3
Solución:
(a-3b) 3 + (3b – c) 3 + (c – a) 3
∵ a – 3b + 3b – c + c – a = 0
∴ (a – 3b) 3 + (3b – c) 3 + (c – a) 3
= 3(a – 3b) (3b – c) (c – a) {∵ a 3 + b 3 + c 3 = 3abc si a + b + c = 0}
Pregunta 13. 2 √2a 3 + 3√3b 3 + c 3 – 3√6abc
Solución:
= (√2a) 3 +(√3b) 3 +c 3 – 3 * √2a * √3b * c
= (√2a + √3b +c)[(√2a) 2 +(√3b) 2 + c 2 – (√2a * √3b) – (√3b * c) – (c * √2a)
= (√2a + √3b +c)(2a 2 + 3b 2 + c 2 – √6ab – √3bc – √2ca)
Pregunta 14. 3 √3a 3 – b 3 – 5√5c 3 – 3√15abc
Solución:
= (√3a) 3 + (-b) 3 +(-√5c) 3 – 3*√3a* (-b) *(-√5c)
= (√3a – b – √5c) [(√3a) 2 + (-b) 2 +(-√5c) 2 – (√3a* -b) – (-b * (-√5c)) – ( -√5c* √3a)
= (√3a – b – √5c)(3a 2 + b 2 + 5c 2 + √3ab – √5bc + √15ca)
Pregunta 15. 2√2 a 3 + 16 √2 b 3 + c 3 – 12abc
Solución:
=(√2a) 3 + (2√2b) 3 + c 3 – (3 * √2a * 2√2b * c)
=(√2a + 2√2b +c) [(√2a) 2 + (2√2b) 2 + c 2 – (√2a* 2√2b) – (2√2b*c) – (c* √2a )
=(√2a + 2√2b +c)[2a 2 + 8b 2 + c 2 – 4ab – 2√2bc – √2ca]
Pregunta 16. Encuentra el valor de x 3 + y 3 – 12xy + 64, cuando x + y = – 4
Solución:
x3 + y3 – 12xy + 64
x + y = -4
Al cubo de ambos lados,
x 3 + y 3 + 3 xy(x + y) = -64
Sustituye el valor de (x + y)
⇒ x 3 + y 3 + 3xy * (-4) = -64
⇒ x 3 + y 3 – 12xy + 64 = 0
Publicación traducida automáticamente
Artículo escrito por DivyansheeVarshney y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA