[20] Factorization

1.(x+1)^2
2.(g+5)^2
3.(y-4)^2
4.(2x-7)^2
5.(6+a)^2
6.(2-3n)^2
7.2(2m-1)^2
8.3(3y+1)^2
9.(x+2)^2
10.(a+6)^2

1. x^2 + 2x +1

=( x + 1 )( x + 1 )
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2. g^2 + 10g + 25

= g^2 + (2)(g)(5) + 5^2

= ( g + 5 )( g + 5 )
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3. y^2 - 8y + 16

= y^2 - (2)(y)(4) + 4^2

= ( y - 4 )( y - 4 )
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4. 4x^2 - 28x + 49

= (2x)^2 - (2)(2x)(7) + 7^2

= ( 2x - 7 )( 2x - 7)
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5. 36 + 12a + a^2

= a^2 + (2)(a)(6) + 6^2

= ( a + 6 )( a + 6 )
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6. 4 - 12n + 9n^2

= 9n^2 - 12n + 4

= (3n)^2 - (2)(3n)(2) + 2^2

= ( 3n - 2 )( 3n - 2)
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7. 8m^2 - 8m + 2

= 2[ 4m^2 - 4m + 1 ] {抽出相同的因數common factor}

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

= 2( 2m - 1 )( 2m - 1)
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8. 27y^2 + 18y + 3

= 3 [ 9y^2 + 6y + 1 ] {抽出相同的因數common factor}

= 3 [ (3y)^2 + (2)(3y)(1) + 1^2 ]

= 3( 3y + 1 )( 3y + 1)
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9. (x+1)^2 + 2(x+1) + 1

其實這題跟第一題也差不多

我們可以Let 這題的(x+1) 做第一題的x
這題就變成: x^2 + 2x +1
=( x + 1 )( x + 1 )

i.e. = [(x+1) + 1][(x+1) + 1]

= ( x + 2 )( x + 2 )
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10. (a+3)^2 + 6(a+3) + 9

= (a+3)^2 + (2)(a+3)(3) + 3^2

= [ (a+3) + 3][ (a+3) + 3]

= ( a + 6 )( a + 6 )
============


其實factorization好簡單,
只要記著以下的formula便可:
1) a^2 + 2ab + b^2 = (a+b)^2

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

3) a^2 - b^2 = (a+b)(a-b)

4) a^3 + b^3 = (a+b)(a^2 - ab + b^2)

5) a^3 - b^3 = (a-b)(a^2 + ab + b^2)

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

2. g^2 + 10g + 25
=g^2 + 2(g)(5) + 5^2
=(g + 5)^2

3. y^2 - 8y + 16
=y^2 - 2(y)(4) + 4^2
=(y - 4)^2

4. 4x^2 - 28x + 49
=(2x)^2 - 2(2x)(7) + 7^2
=(2x - 7)^2

5. 36 + 12a + a^2
=6^2 + 2(6)(a) + a^2
=(6 + a)^2

6. 4 - 12n + 9n^2
=2^2 - 2(2)(3n) + (3n)^2
=(2 - 3n)^2

7. 8m^2 - 8m + 2
=2(4m^2 - 4m + 1)
=2[(2m)^2 - 2(2m)(1) + 1^2]
=2(2m - 1)^2

8. 27y^2 + 18y + 3
=3(9y^2 + 6y + 1)
=3[(3y)^2 + 2(3y)(1) + 1^2]
=3(3y + 1)^2

9. (x+1)^2 + 2(x+1) + 1
=[(x+1) + 1]^2
=(x + 2)^2

10. (a+3)^2 + 6(a+3) + 9
=(a+3)^2 + 2(a+3)(3) + 3^2
=[(a+3) + 3]^2
=(a + 6)^2