因式分解12條...急！！
2017/06/09 19:09
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標題:
因式分解12條...急！！
發問:
1.24+81x^3 2.2x^5y^3-250x^2y^3 3.q^3-343 4.27a^3+b^6 5.a^3-1 6.由此，b^3+b^2-b-1 7.x^2-7x-198 8.由此，*2y+3)^2-7(2y+3)-198 9.6(a-1)b^3+(1-a)b^2 10.4x^2+20xy+25y^2 11.p^2+6pq+9q^2-4x^2 12.9m^2-4n^2+4p^2+12mp
最佳解答:
您好，我是 lop，高興能解答您的問題。 (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)
其他解答:
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)
因式分解12條...急！！
發問:
1.24+81x^3 2.2x^5y^3-250x^2y^3 3.q^3-343 4.27a^3+b^6 5.a^3-1 6.由此，b^3+b^2-b-1 7.x^2-7x-198 8.由此，*2y+3)^2-7(2y+3)-198 9.6(a-1)b^3+(1-a)b^2 10.4x^2+20xy+25y^2 11.p^2+6pq+9q^2-4x^2 12.9m^2-4n^2+4p^2+12mp
最佳解答:
您好，我是 lop，高興能解答您的問題。 (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)
其他解答:
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)
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