问题1.找到以下每个二项式表达式的立方体。
(a)(1 / x + y / 3)
(b)(3 / x-2 / x 2 )
(c)(2x + 3 / x)
(d)(4-1 / 3倍)
解决方案：
(a)(1 / x + y / 3)
Given, (1/x+y/3)3 [(a+b)3=a3+b3+3ab(a+b)]
we know that here, a = 1/x, b= y/3
By using formula:
(1/x+y/3)3 = (1/x)3 + (y/3)3 + 3(1/x)(y/3)(1/x+y/3)
= 1/x3 +y3/27 + y/x(1/x +y/3)
= 1/x3+y3/27 + y/x2 + y2/3x
Hence, (1/x+y/3)3 = 1/x3+y3/27 + y/x2 + y2/3x
(b)(3 / x-2 / x 2 )
Given, (3/x – 2/x2)3 [(a-b)3=a3-b3-3ab(a-b)]
we know that here, a = 3/x, b= y/3
By using formula:
(3/x – 2/x2)3 = (3/x)3 – (2/x2)3 -3(3/x)(2/x2)(3/x -2/x2)
= 27/x3 – 8/x6 – 3(6/x3)(3/x – 2/x2)
= 27/x3 – 8/x6 – 18/x3 (3/x – 2/x2)
= 27/x3 – 8/x6 – 54/x4 – 36/x5
Hence, (3/x – 2/x2)3 = 27/x3 – 8/x6 – 54/x4 – 36/x5
(c)(2x + 3 / x)
Given, (2x+3/x)3 [(a+b)3=a3+b3+3ab(a+b)]
we know that, a = 2x, b = 3/x
By using formula,
(2x+3/x)3 = (2x)3 + (3/x)3 +3(2x)(3/x)(2x+3/x)
= 8x3 + 27/x3 + 18(2x+3/x)
= 8x3 + 27/x3 + 36x + 54/x
Hence, (2x+3/x)2 = 8x3 + 27/x3 + 36x + 54/x
(d)(4-1 / 3倍)
Given, (4-1/3x)3
we know that, a = 4, b = 1/3x [(a-b)3=a3-b3-3ab(a-b)]
By using formula,
(4-1/3x)3 = 43-(1/3x)3 -3(4)(1/3x)(4-1/3x)
= 64 – (1/27x3) – 4/x(4 – 1/3x)
= 64 – (1/27x3) – 16/x + 4/3x
Hence, (4-1/3x) = 64 – (1/27x3) – 16/x + 4/3x
问题2.简化以下各项
(a)(x + 3) 3 +(x-3) 3
(b)(x / 2 + y / 3) 3 –(x / 2-y / 3) 3
(c)(x + 2 / x) 3 +(x – 2 / x) 3
(d)(2x-5y) 3 –(2x + 5y) 3
回答：
(a) (x+3)3 + (x-3)3
The above equation is in the form of a3+b3=(a+b)(a2+b2-ab)
we know that, a = (x+3), b= (x-3)
By using (a3+b3) formula
= (x+3+x-3)[(x+3)3+(x-3)3-(x+3)(x-3)]
= 2x[(x2+32+2*3*x)+(x2+32-2*3*x)-(x2-32)]
= 2x[(x2+9+6x)+(x2+9-6x)-x2+9]
= 2x[x2+27]
= 2x3+54x
Hence, the result of (x+3)3+(x-3)3 is 2x3+54x
(b) (x/2+y/3)3 – (x/2-y/3)3
The above equation is in the form of a3-b3=(a-b)(a2+b2+ab)
we know that, a = (x/2+y/3), b= (x/2-y/3)
By using (a3-b3) formula
= [(x/2+y/3)-(x/2-y/3)][(x/2+y/3)2+(x/2-y/3)2+(x/2+y/3)(x/2-y/3)]
= [2y/3][((x/2)2+(y/3)2+2(x/2)(y/3))+((x/2)2+(y/3)2 -2(x/2)(y/3)+x2/4-y2/9]
= (2y/3)[3x2/4+y2/9]
= x2y/2 +2y3/27
Hence, (x/2+y/3)3 – (x/2-y/3)3 = x2y/2 +2y3/27
(c) (x+2/x)3+(x-2/x)3
The above equation is in the form of a3+b3=(a+b)(a2+b2-ab)
we know that, a = (x+2/x), b= (x-2/x)
By using (a3+b3) formula
= [(x+2/x)+(x-2/x)][ (x+2/x)2+(x-2/x)2 – (x+2/x)(x-2/x)]
= (2x)[((x)2+(2/x)2+2(x)(2/x))+((x)2+(2/x)2-2(x)(2/x)) -(x2-(2/x)2)]
= (2x)[x2+3(2/x)2]
= (2x)[x2 +12/x2]
= 2x3+ 24/x
Hence, (x+2/x)3+(x-2/x)3 = 2x3+ 24/x
(d) (2x-5y)3-(2x+5y)3
The above equation is in the form of a3-b3=(a-b)(a2+b2+ab)
we know that, a = (2x-5y), b= (2x+5y)
By using (a3-b3) formula
= [(2x-5y)-(2x+5y)][(2x-5y)2+(2x+5y)2+(2x-5y)(2x+5y)]
= (-10y)[((2x)2+(5y)2-2(2x)(5y))+((2x)2+(5y)2+2(2x)(5y))+4x2-25y2]
= (-10y)[3(4x2)+25y2]
=(-10y)[12x2+25y2]
= -120x2y – 250y3
Hence, (2x-5y)3-(2x+5y)3 = -120x2y – 250y3
问题3.如果a + b = 10且ab = 21，请找到a 3 + b 3的值。
解决方案：
Given,
a+b = 10, ab = 21
we know that, (a+b)3 = a3+b3+3ab(a+b) ———– 1
substitute a+b = 10, ab = 21 in eq -1
⇒ (10)3 = a3+b3+3(21)(10)
⇒ 1000 = a3+b3+630
⇒ 1000 – 630 = a3+b3
⇒ 370 = a3+b3
Hence, a3+b3=370
问题4.如果ab = 4且ab = 21，则找到3 -b 3的值。
解决方案：
Given,
a-b = 4, ab = 21
we know that, (a-b)3=a3-b3 -3ab(a-b)
Substitute a-b=4, ab= 21 in eq-1
⇒ (4)3 = a3-b3-3(21)(4)
⇒ 64 = a3 -b3-252
⇒ 64+252 = a3 – b3
⇒ 316 = a3-b3
Hence, a3-b3 = 316
问题5.如果(x + 1 / x)= 5，则求出x 3的值+ 1 / x 3
解决方案：
Given, (x+1/x) = 5
we know that, (a+b)3 = a3+b3+3ab(a+b) ————- 1
Substitute (x+1/x) = 5 in eq–1
(x+1/x)3=x3+(1/x)3+3(x)(1/x)(x+1/x)
(5)3 = x3+(1/x)3 + 3(5)
125 -15 = x3+(1/x)3
110 = x3+(1/x)3
Hence, the result is x3+1/x3 = 110
问题6.如果(x-1 / x)= 7，则求出x 3 -1 / x 3的值
解决方案：
Given, If (x-1/x) = 7
we know that, (a-b)3 = a3-b3-3ab(a-b) ————- 1
substitute (x-1/x) = 7 in eq–1
(x-1/x)3 = x3 – 1/x3 -3(x)(1/x)(x-1/x)
(7)3 = x3-1/x3 -3(7)
343+21 = x3 – 1/x3
364 = x3 – 1/x3
Hence, the result is x3-1/x3 = 364
问题7.如果(x-1 / x)= 5，则求出x 3 -1 / x 3的值
解决方案：
Given, If (x-1/x) = 5
we know that, (a-b)3 = a3-b3-3ab(a-b) ————- 1
Substitute (x-1/x) = 5 in eq–1
(x-1/x)3 = x3 – 1/x3 -3(x)(1/x)(x-1/x)
(5)3 = x3-1/x3 -3(5)
125+15 = x3 – 1/x3
x3 – 1/x3=140
Hence, the result is x3-1/x3 = 140
问题8.如果(x 2 + 1 / x 2 )= 51，则找到x 3 -1 / x 3的值。
解决方案：
Given, If (x2+1/x2) = 51
we know that, (a-b)2= a2+b2-2ab ————- 1
substitute (x2+1/x2) = 51 in eq–1
(x-1/x)2 = x2 +1/x2 -2(x)(1/x)
(x-1/x)2 = 51 -2
= 49
x-1/x = ±7
we need to find x3-1/x3
So, a3-b3 = (a-b)(a2+b2+ab)
x3-1/x3 = (x-1/x)(x2+1/x2+x*1/x)
Substitute all known values here,
= 7(51+1)
=7(52)
x3-1/x3 = 364
Hence, the result is x3-1/x3 = 364
问题9.如果(x 2 + 1 / x 2 )= 98，则求x 3 + 1 / x 3的值
解决方案：
Given, (x2+1/x2) = 98
we know that, (x+y)2 = x2+y2+2xy ——– 1
substitute (x2+1/x2) = 98
(x+1/x)2=x2+1/x2+2(x)(1/x)
= 98 + 2
= 100
(x+1/x) = ±10
We need to find x3+1/x3
(x+1/x) = 10 and (x2+1/x2) = 98
x3+1/x3 = 10(98-1)
= 970
Hence, the value of x3+1/x3 = 970
问题10。如果2x + 3y = 13且xy = 6，则找到8x 3 + 27y 3的值
解决方案：
Given, 2x+3y=13, xy=6
We know that,
⇒ (2x+3y)3=133
⇒ 8x3+27y3+3(2x)(3y)(2x+3y) = 2197
⇒ 8x3+27y3+18xy(2x+3y) = 2197
⇒ 8x3+27y3+18xy(2x+3y)= 2197
Substitute the values of 2x+3y=13, xy=6
⇒ 8x3+27y3+18(6)(13)= 2197
⇒ 8x3+27y3 = 2197 – 1404
⇒ 8x3+27y3 = 793
Hence, the value of 8x3+27y3 = 793