分解以下各项：
问题1. a 3 + 8b 3 + 64c 3 – 24abc
解决方案：
We know that
a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
= a3 + 8b3 + 64c3 – 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) (a2 + 4b2 + 16c2 – 2ab – 8bc – 4ca)
问题2. x 3 – 8y 3 + 27z 3 + 18xyz
解决方案：
We can simplify the given equation as :
x3 – 8y3 + 27z3 + 18xyz
= (x)3 + (-2y)3 + (3z)3 – ( 3 * x * (-2y) * (3 z))
= (x – y + 3z) (x2 + 4y2 + 9z2 + 2xy + 6yz – 3zx)
问题3。27x 3 – y 3 – z 3 – 9xyz
解决方案：
27x3 – y3 – z3 – 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) (9x2 + y2 + z2 + 3xy – yz + 3zx)
问题4.(1/27)x 3 – y 3 + 125z 3 + 5xyz
解决方案：
=(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)x2 + y2 +25z2 + (1/3xy) + 5yz – (5/3xz)]
问题5：8x 3 + 27y 3 – 216z 3 + 108xyz
解决方案：
8x3 + 27y3 – 216z3 + 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) (4x2 + 9y2 + 36z2 – 6xy + 18yz + 12zx)
问题6. 125 + 8x 3 – 27y 3 + 90xy
解决方案：
125 + 8X3 – 27y3 + 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 + 4x2 + 9y2– 10x + 6xy + 15y)
问题7：8x 3 – 125y 3 + 180xy + 216
解决方案：
8x3 – 125y3 + 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) (4x2 + 25y2 + 36 + 10xy + 30y – 12x)
问题8：相乘：
(i)x 2 + y 2 + z 2 – xy + xz + yz x + y – z
(ii)x 2 + 4y 2 + z 2 + 2xy + xz – 2yz x- 2y-z
(iii)x 2 + 4y 2 + 2xy – 3x + 6y + 9 x – 2y + 3
(iv)9x 2 + 25y 2 + 15xy + 12x – 20y + 16 x 3x – 5y + 4
解决方案：
(i) (x2 + y2 + z2 – xy + yz + zx) by (x + y – z)
= x3 +y3 – z3 + 3xyz
(ii) (x2 + 4y2 + z2 + 2xy + xz – 2yz) by (x – 2y – z)
= (x -2y-z) [x2 + (-2y)2 + (-z)2 – (x * (- 2y)) – ((-2y)* (z)) – ((-z) (x))]
= x3 + (-2y)3 + (-z)3 – 3x * (-2y) * (-z)
= x3 – 8y3 – z3 – 6xyz
(iii) x2 + 4y2 + 2xy – 3x + 6y + 9 by (x – 2y + 3)
= (x – 2y + 3) (x2 + 4y2 + 9 + 2xy + 6y – 3x)
= (x)3 + (-2y)3 + (3)3 – (3 * x * (-2y) x 3)
= x3 – 8y3 + 27 + 18xy
(iv) 9x2 + 25y2 + 15xy + 12x – 20y + 16 by (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
= 27x3 – 125y3 + 64 + 180xy
问题9.(3x – 2y) 3 +(2y – 4z) 3 +(4z – 3x) 3
解决方案：
(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) {∵ x3 + y3 + z3 = 3xyz if x + y + z = 0}
问题10.(2x – 3y) 3 +(4z – 2x) 3 +(3y – 4z) 3
解决方案：
(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) {∵ x3 + y3 + z3 = 3xyz if x + y + z = 0}
问题11 [[x / 2)+ y +(z / 3)] 3 + [(x / 3)-(2y / 3)+ z] 3 + [(-5x / 6)-(y / 3) -(4z / 3)] 3
解决方案：
[(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)] {∵ a3 + b3 + c3 = 3abc if a + b + c = 0}
问题12.(a – 3b) 3 +(3b – c) 3 +(c – a) 3
解决方案：
(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) {∵ a3 + b3 + c3 = 3abc if a + b + c = 0}
问题13 2√2a3 +3√3b3 + C 3 – 3√6abc
解决方案：
= (√2a)3 +(√3b)3 +c3 – 3 * √2a * √3b * c
= (√2a + √3b +c)[(√2a)2 +(√3b)2 + c2 – (√2a * √3b) – (√3b * c) – (c * √2a)
= (√2a + √3b +c)(2a2 + 3b2 + c2 – √6ab – √3bc – √2ca)
问题14. 3√3a3 – B 3 – 5√5c3 – 3√15abc
解决方案：
= (√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)(3a2 + b2 + 5c2 + √3ab – √5bc + √15ca)
问题15.2√23 + 16√2b 3分配+ C 3 – 12abc
解决方案：
=(√2a)3 + (2√2b)3 + c3 – (3 * √2a * 2√2b * c)
=(√2a + 2√2b +c) [(√2a)2 + (2√2b)2 + c2 – (√2a* 2√2b) – (2√2b*c) – (c* √2a)
=(√2a + 2√2b +c)[2a2 + 8b2 + c2 – 4ab – 2√2bc – √2ca]
问题16。当x + y = – 4时，求出x 3 + y 3 – 12xy + 64的值
解决方案：
x3 + y3 – 12xy + 64
x + y = -4
On Cubing both sides,
x3 + y3 + 3 xy(x + y) = -64
Substitute the value of (x + y)
⇒ x3 + y3 + 3xy * (-4) = -64
⇒ x3 + y3 – 12xy + 64 = 0