因式分解的問題

1. ax*2 + (ab+1)x + b = ax*2 +abx+x+b=x(ax+1) +b(ax+1)= (ax+1)(x+b) 2. x*2 + x +1/4 + yz – z*2 – y*2/4 = .( x*2 + x +1/4) - (z*2 - yz + y*2/4) = .[x*2 + 2x(1/2) +(1/2)*2 ] - [z*2 - 2(y/2)z + (y/2)*2]= (x+ 1/2)*2 - (z - y/2)*2= [(x+ 1/2) + (z - y/2)][(x+ 1/2) - (z - y/2)]= (x - y/2 + z + 1/2)(x + y/2 - z + 1/2) 3. (x + y ) *3 – (x – y ) *3
= [(x + y ) – (x – y )][(x + y ) *2 +(x + y )(x – y ) + (x – y ) *2]= (2y)(x*2+2xy +y*2 +x*2 -y*2 + x*2- 2xy +y*2 )= 2y(3x*2 + y*2) 4. x*3y*3 – 64 = (xy)*3 - 4*3
= (xy - 4)[(xy)*2 +(xy)(4) +4*2]
= (xy - 4)(x*2y*2 + 4xy + 16) 5. 81a*3 + 24b*3 = 3[(3a)*3 + (2b)*2] = 3(3a+2b)[(3a)*2 - (3a)(2b) +(2b)*2]= 3(3a+2b)(9a*2 -6ab +4b*2) 6. a*6 – b*6= (a*2)*3 - (b*2)*3 = [ (a*2) - (b*2)][(a*2)*2 +(a*2)(b*2) + (b*2)*2] = (a+b)(a-b)(a*4 + a*2b*2 + b*4)

1.
ax^2 + (ab+1)x + b = ax^2 + abx + x + b
= ax(x+b) + (x+b) = (ax+1)(x+b)

2.
x^2 + x + (1/4) + yz – z^2 – (y^2/4)
= (1/4)[4x^2 + 4x + 1] - (1/4)[4z^2 + 4yz + y^2]
= (1/4)[2x+1]^2 - (1/4)[2z+y]^2
= (1/4){(2x+1) + (2z+y)} {(2x+1) - (2z+y)}
= (1/4)(2x+y+2z+1) (2x-y-2z+1)

3.
A^3 - B^3 = (A-B) (A^2 + AB + B^2)
(x+y)^3 – (x-y)^3
=(2y) {(x+y)^2 + (x+y)(x-y) + (x-y)^2}
=(2y) {(x^2 +2xy + y^2) + (x^2 - y^2) + (x^2 - 2xy + y^2)}
=(2y) {3x^2 + y^2}

4.
(x^3) (y^3) – 64
= (xy)^3 – 4^3
= (xy – 4) {(xy)^2 + 4xy + 16}

5.
81a^3 + 24b^3
= 3(27a^3 + 8b^3)
= 3{(3a)^3 + (2b)^3}
= 3(3a+2b){9a^2 - 12ab + 4b^2)
= 3 (3a+2b) (3a-2b)^2

6.
a^6 – b^6
= (a^3)^2 – (b^3)^2
= [a^3 + b^3] [a^3 – b^3]
= [(a+b)(a^2 - ab + b^2)] [(a-b)(a^2 + ab + b^2)]
= (a+b)(a-b)(a^2 - ab + b^2) (a^2 + ab + b^2)

2011-11-24 13:06:33 補充：
5.
81a^3 + 24b^3
= 3(27a^3 + 8b^3)
= 3{(3a)^3 + (2b)^3}
= 3(3a+2b){9a^2 - 6ab + 4b^2}

1.
ax^2 + (ab+1)x + b
=ax^2+abx+x+b
=x(ax+1)+b(ax+1)
=(ax+1)(x+b)

2.
x^2 + x +1/4 + yz – z^2 – y^2/4
=4x^2+4x+1+4yz-4z^2-y^2
=(2x+1)^2-4z^2+4yz-y^2
=(2x+1)^2-(4z^2-4yz+y^2)
=(2x+1)^2-(2z-y)^2
=[(2x+1)-(2z-y)][(2x+1)+(2z-y)]
=(2x+y-2z+1)(2x-y+2z+1)

3.
(x + y )^3 – (x – y )^3
=[(x+y)-(x-y)][(x+y)^2+(x+y)(x-y)+(x-y)^2]
=(x+y-x+y)(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2)
=y(3x^2+y^2)
=3x^2y+y^3

4.
x^3y^3 – 64
=(xy)^3-64
=(xy-4)[(xy)^2+4xy+16]
=(xy-4)(xy+4)^2

5.
81a^3 + 24b^3
=3(27a^3+8b^3)
=3(3a+2b)(9a^2+6ab+4b^2)
=3(3a+2b)(3a+2b)^2
=3(3a+2b)^3

6.
a*6 – b*6
=(a-b)(a^5+a^4b+a^3b^2+a^2b^3+ab^4+b^5)

參考資料：
aⁿ-bⁿ=(a-b)(aⁿ⁻¹+aⁿ⁻²b+aⁿ⁻³b²+………+a²bⁿ⁻³+abⁿ⁻²+bⁿ⁻¹)　
aⁿ+bⁿ=(a+b)(aⁿ⁻¹-aⁿ⁻²b+aⁿ⁻³b²-………+a²bⁿ⁻³-abⁿ⁻²+bⁿ⁻¹)　

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