因式分解12條...急！！

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(1)

24+81x^3
= 3(8+27x^3)
= 3[2^3+(3x)^3]
= 3(3x+2)[(3x)^2-2(3x)+2^2]註(1)
= 3(3x+2)(9x^2-6x+4)

(2)

2x^5y^3-250x^2y^3
= 2(x^2)(y^3)(x^3-125)
= 2(x^2)(y^3)(x^3-5^3)
= 2(x^2)(y^3)(x-5)(x^2+5x+5^2)註(2)
= 2(x^2)(y^3)(x-5)(x^2+5x+25)

(3)

q^3-343
= q^3-7^3
= (q-7)(q^2+7q+7^2)註(2)
= (q-7)(q^2+7q+49)

(4)

27a^3+b^6
= (3a)^3 + (b^2)^3
= (3a+b^2)[(b^2)^2-(3a)(b^2)+(3a)^2]註(1)
= (3a+b^2)(9a^2-3ab^2+b^4)

(5)

a^3-1
= a^3-1^3
= (a-1)(a^2+1a+1^2)註(2)
= (a-1)(a^2+a+1)

(6)

b^3+b^2-b-1

代 a = b，
b^3+b^2-b-1
= a^3+a^2-a-1
= (a^3-1)+(a^2-a)
= (a-1)(a^2+a+1) + a(a-1)
= (a-1)(a^2+a+1+a)
= (a-1)(a^2+2a+1)
= (b-1)(b^2+2b+1)

(7)

x^2-7x-198
= x^2 + 11x - 18x - 198
= ( x^2 + 11x ) - ( 18x + 198 )
= x(x+11) - 18(x+11)
= (x-18)(x+11)

(8)

(2y+3)^2-7(2y+3)-198

代 x = 2y + 3，
(2y+3)^2-7(2y+3)-198
= x^2 - 7x - 198
= (x-18)(x+11)
= [(2y+3)-18][(2y+3)+11]
= (2y+3-18)(2y+3+11)
= (2y-15)(2y+14)
= 2(2y-15)(y+7)

(9)

6(a-1)b^3+(1-a)b^2
= 6(a-1)b^3-(a-1)b^2
= (a-1)(b^2)(6b-1)

(10)

4x^2+20xy+25y^2
= ( 4x^2 + 10xy ) + ( 10xy + 25y^2 )
= 2x(2x+5y) + 5y(2x+5y)
= (2x+5y)(2x+5y)
= (2x+5y)^2

(11)

p^2+6pq+9q^2-4x^2
= ( p^2 + 3pq ) + ( 3pq + 9p^2 ) - (2x)^2
= p(p+3q) + 3q(p+3q) - (2x)^2
= (p+3q)(p+3q)- (2x)^2
= (p+3q)^2 - (2x)^2
= (p+3q+3x)(p+3q-2x)註(3)

(12)

9m^2-4n^2+4p^2+12mp
= 9m^2 + 12mp + 4p^2 - 4n^2
= ( 9m^2 + 6mp ) + ( 6mp + 4p^2 ) - (2n)^2
= 3m(3m+2p) + 2p(3m+2p) - (2n)^2
= (3m+2p)(3m+2p) - (2n)^2
= (3m+2p)^2 - (2n)^2
= (3m+2p+2n)(3m+2p-2n)註(3)

註：

(1) 緊記公式 x^3 + y^3 = (x+y)(x^2-xy+y^2)
(2) 緊記公式 x^3 - y^3 = (x-y)(x^2+xy+y^2)
(3) 緊記公式 x^2 - y^2 = (x+y)(x-y)

1.24+81x^3
=3(2^3+x^3)
=3(2+x)(2^2-2x+x^2)
=3(2+x)(4-2x+x^2)

2. 2x^5y^3-250x^2y^3
=2x^2y^3(x^3-5^3)
=2x^2y^3(x-5)(x^2+5x+25)

3.q^3-343
=q^3-7^3
=(q-7)(q^2+7q+49)

4.27a^3+b^6
=(3a)^3+(b^2)^3
=(3a+b^2)(9a^2-3ab^2+b^4)

5.a^3-1
=a^3-1^3
=(a-1)(a^2+a+1)
6.b^3+b^2-b-1
=b^3-1+(b^2-b)
=(b-1)(b^2+b+1)+b(b-1)
=(b-1)(b^2+2b+1)
=(b-1)(b+1)^2

7.x^2-7x-198
=(x+11)(x-18)

8.(2y+3)^2-7(2y+3)-198
=(2y+3+11)(2y+3-18)
=(2y+14)(2y-15)

9.6(a-1)b^3+(1-a)b^2
=6(a-1)b^3-(a-1)b^2
=b^2(a-1)(6b-1)

10.4x^2+20xy+25y^2
=(2x+5y)^2

11.p^2+6pq+9q^2-4x^2
=(p+3q)^2-(2x)^2
=(p+3q-2x)(p+3q+2x)

12.9m^2-4n^2+4p^2+12mp
=(3m+2p)^2-(2n)^2
=(3m+2p+2n)(3m+2p-2n)