Question

分解因式 a^2(b-c)+b^2(c-a)+c^2(a-b)

Answer 1

经典的一个因式分解,提供一个思路：把后面两个式子拆开,凑(b-c)
原式= a^2(b-c)+(b^2)c-(c^2)b+a(c^2)-a(b^2)
= a^2(b-c)+bc(b-c)-a(b+c)(b-c)
=(b-c)[(a^2)-ab-ac+bc]
=(b-c)(a-b)(a-c)

Answer 2

a^2(b-c)+b^2(c-a)+c^2(a-b)
= a^2(b-c)+b^2c-ab^2+ac^2-bc^2
= a^2(b-c)+b^2c-bc^2+ac^2-ab^2
=a^2(b-c)+bc(b-c)+a(c+b)(c-b)
=(b-c)(a^2+bc-ac-ab)
=(b-c)[a(a-b)-c(a-b)]
=(b-c)(a-b)(a-c)

Answer 3

原式=ba^2-ca^2+cb^2-ab^2+c^2[a-b]
=(ba^2-ab^2)-(ca^2-cb^2)+c^2[a-b]
=ab(a-b)-c(a^2-b^2)+c^2[a-b]
=ab(a-b)-c(a-b)(a+b)+c^2[a-b]
=(a-b)(ab-ac-bc+c^2)
=(a-b)[(ab-ac)-(bc-c^2)]
=(a-b)[a(b-c)-c(b-c)]
=(a-b)[(a-c)(b-c)]
=(a-b)(a-c)(b-c)

Answer 4

原式=a^2b-ab^2-a^2c+b^2c+c^2(a-b)
=ab(a-b)-c(a^2-b^2)+c^2(a-b)
=ab(a-b)-c(a-b)(a+b)+c^2(a-b)
=(a-b)(ab-ac-bc+c^2)
=(a-b)[a(b-c)-c(b-c)]
=(a-b)(a-c)(b-c)

Answer 5

a^2(b-c)+b^2(c-a)+c^2(a-b)
=a^2b-a^2c+b^2(c-a)+c^2a-c^2b
=a^2b-c^2b+b^2(c-a)+c^2a-a^2c
=b(a^2-c^2)+b^2(c-a)+ca(c-a)
=-b(c+a)(c-a)+b^2(c-a)+ca(c-a)
=(c-a)(-bc-ab+b^2+ac)
=(c-a)[(b^2-bc)+(ac-ab)]
=(c-a)[b(b-c)-a(b-c)]
=(c-a)(b-c)(b-a)