f.2-f.3因式分解(15分)

應用公式：
(u + v)² = u² + 2uv + v²
(u - v)² = u² - 2uv + v²
u² - v² = (u + v)(u - v)
u³ + v³ = (u + v)(u² - uv + v²)
u³ - v³ = (u - v)(u² + uv + v²)


1.)
x² + 6x + 9
= x² + 2x(3) + (3)²
= (x + 3)² ... (答案)


2.)
題目中的 -25x 應是 -26x
x² - 26x + 169
= x² - 2x(13) + (13)²
= (x - 13)² ... (答案)


3.)
x² - 64
= x² - 8²
= (x + 8)(x - 8) ... (答案)


4.)
81x² - 25
= (9x)² - 5²
= (9x + 5)(9x - 5) ... (答案)


5.)
4x² - 25y²
= (2x)² - (5y)²
= (2x + 5y)(2x - 5y) ... (答案)


6.)
x² + 7x + 10
= x² + 2x + 5x + 10
= (x² + 2x) + (5x + 10)
= x(x + 2) + 5(x + 2)
= (x + 5)(x + 2) ... (答案)


7.)
x³ + 64
= x³ + 4³
= (x + 4)(x² - 4x + 4²)
= (x + 4)(x² - 4x + 16) ... (答案)


8.)
512 - a³
= 8³ - a³
= (8 - a)(8² + 8a + a²)
= (8 - a)(64 + 8a + a²) ... (答案)


9.)
3x² + 12x + 12
= 3(x2 + 4x + 4)
= 3[x² + 2x(2) + (2)²]
= 3(x + 2)² ... (答案)


10.)
題目中的 -24xy 應是 -42xy。
9x² - 42xy + 49y²
= (3x)² - 2(3x)(7y) + (7y)²
= (3x - 7y)² ... (答案)


11.)
設 u = 2x - 1
(2x - 1)² - 25
= u² - 5²
= (u + 5)(u - 5)
= (2x - 1 + 5)(2x - 1 - 5)
= (2x + 4)(2x - 6)
= 2(x + 2)2(x - 3)
= 4(x + 2)(x - 3) ... (答案)


12.)
50 - 8y²
= 2(25 - 4y²)
= 2[5² - (2y)²]
= 2(5 + 2y)(5 - 2y) ... (答案)


13.)
8x² - 38x + 35
= 8x² - 10x - 28x + 35
= (8x² - 10x) - (28x - 35)
= 2x(4x - 5) - 7(4x - 5)
= (4x - 5)(2x - 7) ... (答案)


14.)
20 - 7x - 6x²
= 20 - 15x + 8x - 6x²
= (20 - 15x) + (8x - 6x²)
= 5(4 - 3x) + 2x(4 - 3x)
= (5 + 2x)(4 - 3x) ... (答案)


15.)
題目中的 -8y² 應為 -8y³。
-343 - 8y³
= -(343 + 8y³)
= -[7³ + (2y)³]
= -(7 + 2y)[7² - 7(2y) + (2y)²]
= -(7 + 2y)(49 - 14y + 2y²) ... (答案)


16.)
(x - y)³ - 8(x - y)³
= (x - y)³(1 - 8)
= -7(x - y)³ ... (答案)
= 7(y - x)³ ... (答案)


17.)
x² - 2xy + y²
= (x - y)² ... (答案)


18.)
設 u = x + 1
x² - 2x(x + 1) + (x+1)²
= x² - 2xu + u²
= (x - u)²
= [x - (x + 1)]²
= (-1)²
= 1 ... (答案)

1.)x^2+6x+9
=(x+3)^2//
2.)x^2-26x+169(你是否將26打錯成25呀?)
=(x-13)^2//
3.)x^3-64(你是否將x^3打錯成x^2呀?)
=(x-4)(x^2+8x+16)//
4.)81x^2-25
=(9x)^2-(5)^2
=(9x-5)(9x+5)
5.)4x^2-25y^2
=(2x)^2-(5y)^2
=(2x-5y)(2x+5y)
6.)x^2+7x+10
=(x+5)(x+2)//
7.)x^3+64
=(x+4)(x^2-8x+16)//
8.)512-a^3
=(8-a)(64+16a+a^2)//
9.)3x^2+12x+12
=3(x^2+4x+4)
=3(x+2)^2//
10.)9x^2-42xy+49y^2(你是否將42打錯成24呀?)
=(3x)^2-42xy+(7y)^2
=(3x-7y)^2//
11.)(2x-1)^2-25
=(2x-1)^2-(5)^2
=(2x-1-5)(2x-1+5)
=(2x-6)(2x+4)//
12.)50-8y^2
=2(25-4y^2)
=2(5-2y)(5+2y)//
13.)8x^2-38x+35
=(4x-5)(2x-7)
14.)20-7x-6x^2
=(4-3x)(5+2x)//
15.)-343-8y^3(你是否將x^3打錯成x^2呀?)
=-[(7)^3-(2y)^3]
=-(7-2y)(49+28y+4y^2)
=(2y-7)(49+28y+4y^2)//
16.) (x-y)^3-8(x-y)^3
=[(x-y)-2(x-y)][(x-y)^2+4(x-y)^2+4(x-y)^2]
=(y-x)[x^2-2xy+y^2+8(x^2-2xy+y^2)]
=(y-x)(9x^2-18xy+9y^2)//
17.)x^2-2xy+y^2
=(x-y)^2//
18.) 化簡:
x^2-2x(x+1)+(x+1)^2
設(x+1)=y,
x^2-2x(y)+(y)^2
=(x-y)^2
=[x-(x+1)]^2
=1//