F.2 Factorize

Factorize the following.

1). 4 ( a + 2b ) ² - ( a - b ) ²

＝（２ａ＋４ｂ）² －（ａ－ｂ）²

＝〔（２ａ＋４ｂ）＋（ａ－ｂ）〕〔（２ａ＋４ｂ） －（ａ－ｂ）〕

＝（３ａ＋３ｂ）（ａ＋５ｂ）＃

＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
Simplify the following.

2). x ² - xy + 3x - 3y
________________
2x ² + 2xy + 6x +6y

　ｘ（ｘ－ｙ）＋３（ｘ－ｙ）
＝－－－－－－－－－－－－－
　２ｘ（ｘ＋ｙ）＋６（ｘ＋ｙ）

　（ｘ－ｙ）（ｘ＋３）
＝－－－－－－－－－－
　（ｘ＋ｙ）（２ｘ＋６）

　（ｘ－ｙ）（ｘ＋３）
＝－－－－－－－－－－－
　２（ｘ＋ｙ）（ｘ＋３）

　　（ｘ－ｙ）
＝－－－－－－
　２（ｘ＋ｙ）　＃

＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
3). (2m + 5 ) ÷ (5 + 2m ) ² over ( 3 - n ) ² × 2 m + 5 over 3 m- mn

＝１／(2m + 5 )　／　( 3 - n ) ² × 2 m + 5 ／ ｍ（3 - n ）

＝１／(2m + 5 )　／　( 3 - n ) × （2 m + 5）／ｍ

＝１／(2m + 5 )　×　ｍ／( 3 - n ) × （2 m + 5）

＝ｍ／（３－ｎ）＃

－－－－－－－－－－－－－－－－　ｆｉｎｉｓｈ　－－－－－－－－－－－－－－－

1) 4 ( a + 2b ) ² - ( a - b ) ²
= 4(a² + 4ab + 4b²) - (a² - 2ab + b²)
= 4a² + 16ab + 16b² - a² + 2ab - b²
= 3a² + 18ab + 15b²
= 3(a² + 6ab + 5b²)
= 3(a + b)(a + 5b)


2). x ² - xy + 3x - 3y
________________
2x ² + 2xy + 6x +6y

= x(x - y) + 3(x - y)
________________
2x(x + y) + 6(x +y)

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

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


(3). (2m + 5) ÷ (5 + 2m) ² / (3 - n) ² × 2 m + 5/3 m- mn
= (2m + 5) × [(3 - n)² /(5 + 2m)²] × 2 m + 5/3 m- mn
= [(4m²+10m)(9 - 6n + n²) + (5 + 2m)²(5/3 m- mn)]/(5 + 2m)²
=[36m² + 90mn - 24m²n - 60mn + 4m²n² + 10mn² + 125m/3 - 25mn + 100m²/3 - 20m²n +20m^3/3 - 4m^3n²]/(5 + 2m)²
= [8m²/3 + 5mn - 44m²n + 4m²n² + 10mn² + 125m/3 + 20m^3/3 - 4m^3n²]/(5 + 2m)²
=m[8m + 15n - 132mn + 12mn² + 30n² + 125 + 20m² - 4m²n²]/[3(5 + 2m)²]

Factorize the following.
1). 4( a + 2b )² - ( a - b )²
這題用平方差的公式作因式分解x2 – y2 = (x+y)(x-y)
4( a + 2b )² - ( a - b )²
= [2( a + 2b )]² - ( a - b )²
= [2( a + 2b ) - ( a - b )][2( a + 2b ) + ( a - b )]
= [2a + 4b - a + b ][2a + 4b + a - b]
= (a + 5b)(3a + 3b)
= 3(a + 5b)(a + b)

Simplify the following.
2). x^² - xy + 3x - 3y
─────────
2x^² + 2xy + 6x +6y

x(x – y) + 3(x – y)
=─────────
2x(x + y) + 6(x +y)

(x + 3)(x – y)
=───────
(2x + 6)(x + y)

(x + 3)(x – y)
=───────
2(x + 3)(x + y)

(x – y)
=─────
2(x + y)

(3). [(2m + 5 ) ÷ (5 + 2m )²] / ( 3 - n ) ² × (2 m + 5)/(3m- mn)
= (5 + 2m) ÷ (5 + 2m ) ² / ( 3 - n ) ² × (2 m + 5)/(3m- mn)
= 1 / [(5 + 2m )( 3 - n )²] × (2 m + 5)/(3 m- mn)
= 1 / [( 3 - n )²(3m- mn)]
= 1/[m( 3 - n )²(3- n)]

1, 4 ( a + 2b ) ² - ( a - b ) ²
= [2 ( a + 2b ) - ( a - b )][2 ( a + 2b ) + ( a - b )]
= [ a + 5b] [ 3a +3b]
= 3 [ a + 5b] [ a +b]

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

2006-11-25 11:34:10 補充：
You may consider the identitya² - b² = (a b)(a-b)

2006-11-25 11:34:52 補充：
Sorry. It should be a² - b² = (a b)(a-b)

2006-11-25 11:37:37 補充：
I apology for the fact that the operator "plus" cannot display properly