中二數---唔該有心人教下我6題數(20點)...

1. 把 4(x + y) – 6a(x + y)² 分解為因式

4(x+y) - 5a(x+y)²

= (x+y)[4 + 5a(x+y)]

= (x+y)(4 + 5ax + 5ay)

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2. 把 9(a – b)² + 12b(b – a) 分解為因式

9(a-b)² + 12b(b-a)

= 3[3(a-b)](a-b) - 4b[3(a-b)]

= 3(a-b)[3(a-b) - 4b]

= 3(a-b)(3a - 3b - 4b)

= 3(a-b)(3a - 7b)

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3. 把 ab² + b – ab - 1分解為因式

ab² + b - ab - 1

= (ab² - ab ) + (b - 1)

= ab(b - 1) + (b - 1)

= (ab + 1)(b - 1)

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4. 把 27d² – 48e²分解為因式

27d² - 48e²

= 3(9d²) - 3(16e²)

= 3[(3d)² - (4e)²]

= 3(3d - 4e)(3d + 4e) 【根據 a²-b²=(a-b)(a+b)】

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5. 把下列分解為因式
(a) (a + b)² – 144

(a+b)² - 144
= (a+b)² - 12²
= [(a+b)-12][(a+b)+12]【根據 a²-b²=(a-b)(a+b)】
= (a+b-12)(a+b+12)

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(b) 1 – (s – t)²

1 - (s-t)²
= 1² - (s-t)²
= [1-(s-t)][1+(s-t)]【根據 a²-b²=(a-b)(a+b)】
= (1-s+t)(1+s-t)

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(c) 3(m – n)² – 27

3(m-n)² - 27
= 3[(m-n)² - 9]
= 3[(m-n)² - 3²]
= 3[(m-n)-3][(m-n)+3]【根據 a²-b²=(a-b)(a+b)】
= 3(m-n-3)(m-n+3)

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(d) 80 – 45(x + y)²

80 - 45(x+y)²
= 5[16 - 9(x+y)²]
= 5[4² - (3(x+y))²]
= 5[4 - 3(x+y)][4 + 3(x+y)]【根據 a²-b²=(a-b)(a+b)】
= 5(4 - 3x - 3y)(4 + 3x + 3y)

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(e) 2(p + 3)² – 72

2(p+3)² - 72
= 2[(p+3)² - 36]
= 2[(p+3)² - 6²]
= 2[(p+3)-6][(p+3)+6]【根據 a²-b²=(a-b)(a+b)】
= 2(p-3)(p+9)

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(f)125 – 5(5 – k)²

125 - 5(5-k)²
= 5[25 - (5-k)²]
= 5[5² - (5-k)²]
= 5[5 - (5-k)][5 + (5-k)]【根據 a²-b²=(a-b)(a+b)】
= 5(k)(10-k)

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6. 求下列多項式的L.C.M.
(a) 3x ， 4y

3x = 3 × x
4y = 4 × y

LCM = 3 × 4 × x × y
LCM = 12xy

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(b) 2a ， 2a – 3

2a = 2 × a
2a-3 = 2a-3

LCM = 2 × a × (2a-3)
LCM = 2a(2a-3)
LCM = 4a² - 6a

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(c) 4m – 2n ， 6(2m - n)

4m-2n = 2 × (2m-n)
6(2m-n) = 2 × 3 × (2m-n)

LCM = 2 × 3 × (2m-n)
LCM = 6(2m-n)
LCM = 12m - 6n

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(d) 5p + 10q ， 3p + 6q

5p + 10q = 5 × (p + 2q)
3p + 6q = 3 × (p + 2q)

LCM = 3 × 5 × (p + 2q)
LCM = 15(p+2q)
LCM = 15p + 30q

1. 把 4(x + y) – 6a(x + y)2 分解為因式

4 ( x + y ) = 2 [ 2 ( x + y ) ]
6a ( x + y ) ^ 2 = 3a ( x + y ) [ 2 ( x + y ) ]
∴ 4 ( x + y ) - 6a ( x + y ) ^2 = 2 ( x + y ) [ 2 - 3a ( x + y ) ]


2. 把 9(a – b)2 + 12b(b – a) 分解為因式

9 ( a - b )^2 = 3 ( a - b ) [ 3 ( a - b) ]
12b ( b - a ) = -12b ( a - b )

= 3 ( a - b ) ( -4b )

∴9 ( a - b )^2 + 12b ( b - a ) = 9 ( a - b )^2 - 12b ( a - b )




= 3 ( a - b ) [ 3 ( a - b ) - 4b ]


3. 把 ab2 + b – ab - 1分解為因式

ab^2 + b - ab - 1 = ab^2 - ab + b - 1


= ( ab^2 - ab ) + ( b - 1 )
= ab ( b - 1 ) + ( b - 1 )
= ( ab + 1 ) ( b - 1 )


4. 把 27d2 – 48e2分解為因式

27d^2 - 48 e^2 = 3 ( 9d^2 - 16e^2 )


= 3 [ ( 3d )^2 - ( 4e )^2 ]
= 3 ( 3d + 4e ) ( 3d - 4e ) ...... a^2 - b^2 = ( a + b ) ( a - b )


5. 把下列分解為因式 ...... 以下 a - f 各小題都是以 a^2 - b^2 = ( a + b ) ( a - b ) 的公式為基礎
(a) ( a + b )^2–144 = ( a + b ) ^2 - 12^2


= ( a + b + 12 ) ( a - b - 12 )

(b) 1–( s – t )^2 = 1^2 - ( s - t )^2


= ( 1 - s - t ) ( 1 + s + t )

(c) 3 ( m - n )^2–27 = 3 [ ( m - n )^2 - 9 ]


= 3 [ ( m - n )^2 - 3^2 ]
= 3 ( m - n - 3 ) ( m - n + 3 )

(d) 80–45 ( x + y )^2 = 5 [ 16 - 9 ( x + y )^2 ]


= 5 { 4^2 - [ 3 ( x + y ) ]^2 }
= 5 [ 4 - 3 ( x + y ) ] [ 4 + 3 ( x + y) ]
= 5 ( 4 - 3x - 3y ) ( 4 + 3x + 3y )

(e) 2 ( p + 3 )^2–72 = 2 [ ( p + 3 )^2 - 36 ]


= 2 [ ( p + 3 )^2 - 6^2 ]
= 2 ( p + 3 + 6 ) ( p + 3 - 6 )
= 2 ( p + 9 ) ( p - 3 )

(f) 125 - 5 ( 5–k )^2 = 5 [ 25 - ( 5 - k )^2 ]


= 5 [ 5^2 - ( 5 - k )^2 ]
= 5 [ 5 + ( 5 - k ) ] [ 5 - ( 5 - k ) ]
= 5 ( 5 + 5 - k ) ( 5 - 5 + k )
= 5 ( 10 - k ) k
= 5k ( 10 - k )

6. 求下列多項式的L.C.M.
(a) 3x ， 4y
L.C.M. = 3x ( 4y )

= 12 xy

(b) 2a ， 2a - 3
L.C.M. = 2a ( 2a - 3 )

= 4a^2 - 6a

(c) 4m–2n ， 6 ( 2m - n )

4m - 2n = 2 ( 2m - n )
6 ( 2m - n ) = 3 [ 2 ( 2m - n ) ]

L.C.M. = 3 [ 2 ( 2m - n ) ]

= 6 ( 2m - n )
= 12m - 6n

(d) 5p + 10q ， 3p + 6q

5p + 10q = 5 ( p + 2q )
3p + 6q = 3 ( p + 2q )

L.C.M. = 5 ( p + 2q ) ( 3 )

= 15 ( p + 2q )
= 15p + 30q

1
4(x + y) – 6a(x + y)^2
=2(x+y)[2-3a(x+y)]

2
9(a - b)^2 + 12b(b - a)
=9(a - b)^2 - 12b(a - b)
=3(a - b)[3 ( a - b ) - 4b]
=3(a - b)(3a - 7b)

3
ab^2 + b - ab - 1
=b(ab + 1) - (ab + 1)
=(ab + 1)(b - 1)

4
27d^2 – 48e^2
=3 ( 9d^2 - 16e^2)
=3 [ ( 3d )^2 - ( 4e )^2 ]
=3 (3d - 4e)(3d + 4e)

5a
(a + b)^2 - 144
=(a + b)^2 - 12^2
=(a + b + 12)(a + b - 12)

5b
1 - (s - t)^2
=1^2 - (s - t)^2
=(1 - s + t)(1 + s - t)

5c
3(m – n)^2 – 27
=3 [ (m - n)^2 - 3^2 ]
=3(m - n - 3)(m - n + 3)

5d
80 – 45(x + y) 2
=5 { 4^2 - [3(x + y)]^2 }
=5 [4 - 3(x + y)][4 + 3(x + y)]

5e
2(p + 3)^2 – 72
=2 [ (p + 3)^2 - 6^2]
=2(p + 3 + 6)(p + 3 - 6)
=2(p + 9)(p - 3)

5f
125 – 5(5 – k)^2
=5 [ 5^2 - (5 - k)^2]
=5 (5 - 5 + k)(5 + 5 - k)
=5k(10 - k)

6a
L.C.M. = 12xy

6b
L.C.M. = 2a(2a - 3)

6c
4m - 2n = 2(2m - n)
L.C.M. = 6(2m - n)

6d
5p + 10q = 5(p +2q)
3p + 6q = 3(p + 2q)
L.C.M. = 15(p + 2q)

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