急~恆等式與因式分解~(40點)~!!!

您好，我是 lop****** ，高興能解答您的問題。

問 1：展開 (6a-5b)^2-(a-2b)^2

解 1：

(6a-5b)^2-(a-2b)^2

= 6a(6a-5b) - 5b(6a-5b) - [ a(a-2b) - 2b(a-2b) ]

= 36a^2- 30ab - 30ab + 25b^2- (a^2 - 2ab - 2ab + 4b^2 )

= 36a^2 - 60ab + 25b^2 - a^2 + 2ab + 2ab - 4b^2

= 35a^2 + 21b^2 - 56ab

答 1：(6a-5b)^2-(a-2b)^2 = 35a^2 + 21b^2 - 56ab

問 2：展開 (x+1)^2(x-1)^2

解 2：

(x+1)^2(x-1)^2

= [(x+1)(x-1)]^2

= [x(x-1)+1(x-1)]^2

= (x^2 - x + x - 1)^2

= (x^2 - 1)^2

= x^2 (x^2 - 1) - 1(x^2 - 1)

= x^4- x^2 - ( x^2 - 1 )

= x^4 - x^2 - x^2 + 1

= x^4 - 2x^2 + 1

答 2：(x+1)^2(x-1)^2 = x^4 - 2x^@ + 1

問 3：因式分解 3x^6-48x^4+192x^2

解 3：

3x^6-48x^4+192x^2

= 3x^2 ( x^4 - 16x^2 + 64 )

= 3x^2 ( x^4 - 8x^2 - 8x^2 + 64 )

= 3x^2 [ x^2 ( x^2 - 8 ) - 8 ( x^2 - 8 )]

= 3x^2 [ ( x^2 - 8 )( x^2 - 8 ) ]

= 3x^2 ( x^2 - 8 )^2

答 3：3x^6-48x^4+192x^2 = 3x^2 ( x^2 - 8 )^2
問 4：因式分解 rx-ry+ty-tx

解 4：

rx - ry + ty - tx

= r (x-y) - t (x-y)

= (r-t)(x-y)

答 4：rx - ry + ty - tx = (r-t)(x-y)

問 5：因式分解 (x+y)(2x-y)-3x-3y

解 5：

(x+y)(2x-y)-3x-3y

= (x+y)(2x-y) - 3(x+y)

= (x+y)(2x-y-3)

答 5：(x+y)(2x-y)-3x-3y = (x+y)(2x-y-3)

問 6：因式分解 (x+y)(2a-3b)+(x+y)(2b-c)

解 6：

(x+y)(2a-3b)+(x+y)(2b-c)

= (x+y)(2a-3b+2b-c)

= (x+y)(2a-b-c)

答 6：(x+y)(2a-3b)+(x+y)(2b-c) = (x+y)(2a-b-c)

問 7：因式分解 27-x^3

解 7：

27-x^3

= 3x^2 - x^3 + 9x - 3x^2 + 27 - 9x

= x^2 (3-x) + 3x(3-x) + 9(3-x)

= (3-x)(x^2+3x+9)

答 7：27-x^3 = (3-x)(x^2+3x+9)

問 8：因式分解 16m^3+2

解 8：

16m^3+2

= 2(8m^3 + 1)

= 2(8m^3 + 4m^2 - 4m^2 - 2m + 2m + 1 )

= 2[4m^2 (2m+1) - 2m(2m+1) + 1(2m+1)]

= 2(2m+1)(4m^2-2m+1)

答 8：16m^3+2 = 2(2m+1)(4m^2-2m+1)



2011-07-15 16:27:04 補充：
You mean ...

1. (6a-5b)^2-(a-2b)^2
=36a^2 -25b^2 -a^2 -4b^2
=35a^2 -21b^2

2. (x+1)^2(x-1)^2
=(x^2 +2x +1)(x^2 -2x +1)
=x^4 -2x^3 +x^2 +2x^3 -4x^2 +2x +x^2 -2x +1
=x^4 +2x^2 -4x^2 +1
=x^4 -2x^2 +1

3. 3x^6-48x^4+192x^2
=3x^2 (x^4-16x^2+64)
=3x^2 (x^2-8)^2

4. rx-ry-tx+ty
=(rx-ry)-(tx+ty)
=r(x-y)-t(x-y)
=(x-y)(r-t)

5. (x+y)(2x-y)-3x-3y
=(x+y)(2x-y)-(3x+3y)
=(x+y)(2x-y)-3(x+y)
=(x+y)(2x-y-3)

6. (x+y)(2a-3b)+(x+y)(2b-c)
=(x+y)[(2a-3b)+(2b-c)]
=(x+y)(2a-3b+2b-c)
=(x+y)(2a-b-c)

7. 27-x^3
=(3-x)(3^2+3x+x^2)
=(3-x)(9+3x+x^2)
=(3-x)(x^2+3x+9)

8.16m^3 +2
=2(8m^3+1)
=2(3m+1)(9m^2 -3m +1)

1. (6a-5b)^2-(a-2b)^2
=(6a)^2-2(6a)(5b)+(5b)^2-(a^2-2(a)(2b)+(2b)^2)
=36a^2-60ab+25b^2-a^2+4ab-4b^2
=35a^2-56ab+21b^2
2. (x+1)^2(x-1)^2
=((x+1)(x-1))^2
=(x^2-1^2)^2
=(x^2-1)^2
=(x^2)^2-2(x^2)(1)+1^2
=x^4-2x^2+1
3. 3x^6-48x^4+192x^2
=3x^2(x^4-16x^2+64)
=3x^2((x^2)^2-2(x^2)(8)+8^2)
=3x^2(x^2-8)^2
4. rx-ry+ty-tx
=r(x-y)+t(y-x)
=r(x-y)-t(x-y)
=(r-t)(x-y)
5. (x+y)(2x-y)-3x-3y
=(x+y)(2x-y)-3(x+y)
=(x+y)(2x-y-3)
6. (x+y)(2a-3b)+(x+y)(2b-c)
=(x+y)(2a-3b+2b-c)
=(x+y)(2a-b-c)
7. 27-x^3
=3^3-x^3
=(3-x)(3^2+3x+x^2)
=(3-x)(9+3x+x^2)
8. 16m^3+2
=2(8m^3+1)
=2((2m)^3+1^3)
=2(2m+1)((2m)^2-(2m)(1)+1^2)
=2(2m+1)(4m^2-2m+1)

Expand:

1. (6a-5b)² - (a-2b)²
= [(6a)² - 2(6a)(5b) + (5b)²] - [a² - 2a(2b) + (2b)²]
= (36a² - 60ab + 25b²) - (a² - 4ab) + 4b²)
= 36a² - a² - 60ab + 4ab +25b² - 4b²
= 35a² - 56ab + 21b²

2. (x+1)²(x-1)²
= [(x+1)(x-1)]²
= (x² - 1)²
= (x²)² - 2(x²)(1) +1²
= x^4 - 2x² + 1


Factorize:
3. 3x^6 - 48x^4 + 192x²
= 3x² (x^4 - 16x² + 64)
= 3x² [ (x²)² - 2(x²)(8) + 8² ]
= 3x² [ (x² - 8)² ]
= 3x²(x-8)²

4. rx-ry+ty-tx
= r (x-y) + ty - tx
= r (x-y) - t (x-y)
= (r-t)(x-y)

5. (x+y)(2x-y)-3x-3y
= (x+y)(2x-y) -3(x+y)
= (x+y) [ (2x-y) - 3 ]
= (x+y) ( 2x-y-3 )
= (x+y)(2x-y-3)

6. (x+y)(2a-3b)+(x+y)(2b-c)
= (x+y) [ (2a-3b) + (2b-c) ]
= (x+y) ( 2a-3b+2b-c )
= (x+y)(2a-b-c)

7. 27 - x³
= 3³ - x³
= (3-x) [ 3² + (3)(x) + x² ]
= (3-x) ( 9+3x+x² )

8. 16m³ + 2
= 2 (8m³ + 1)
= 2 [ (2m)³ + 1 ]
= 2 { [(2m)+1] [ (2m)² - (2m)(1) + 1² ] }
= 2 [ (2m+1) ( 4m² - 2m + 1 ) ]
= 2 (2m+1) (4m²-2m+1)

2011-07-15 17:03:15 補充：
Q3 最後一行打錯了，sry ~~
應該係「= 3x²(x²-8)² 」，而唔係「= 3x²(x-8)² 」

2011-07-15 17:05:32 補充：
所以如果你學了開方的話，之後仲有幾個steps 的...

= 3x² (x²-8)²
= 3x² [(x+√8)(x-√8)]²
= 3x² (x+√8)² (x-√8)²

Using identity to solve the questions!

2011-07-16 14:57:31 補充：
(x+1)^2(x-1)^2
(x+1)(x-1)(x+1)(x-1)
(x^2-1)^2
x^4-2x^2+1
____________________