Phân tích đa thức thành nhân tử:
a, \(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right);\)
b,\(a^3\left(c-b^2\right)+b^3\left(c-a^2\right)+c^3\left(a-b^2\right).\)
K
Khách
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S
4 tháng 10 2018
Đặt A là tên biểu thức; \(a+b-c=x;b+c-a=y;c+a-b=z\)
Khi đó \(x+y+z=a+b-c+b+c-a+c+a-b=a+b+c\)
=>\(A=\left(x+y+z\right)^3-x^3-y^3-z^3=\left[\left(x+y\right)+z\right]^3-x^3-y^3-z^3\)
\(=\left(x+y\right)^3+z^3+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\)
\(=x^3+y^3+3xy\left(x+y\right)+z^3+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\)
\(=3\left(x+y\right)\left(xy+xz+yz+z^2\right)\)
\(=3\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]=3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)
\(=3\left(a+b-c+b+c-a\right)\left(b+c-a+c+a-b\right)\left(c+a-b+a+b-c\right)\)
\(=3.2b.2c.2a=24abc\)
SM
4 tháng 9 2018
A = ( a + b + c )3 + ( a - b - c )3 + ( b - c - a )3 + ( c - a - b )3
= [ ( a + b ) + c ]3 + [ ( a - b ) - c ]3 + [ ( - c ) - ( a - b ) ] 3 + [ c - ( a + b ) ]3
= ( a + b )3 + 3.( a + b )2.c + 3.( a + b ).c2 + c3 + ( a - b )3 - 3.( a - b )2.c + 3.( a - b ).c2 - c3 + ( - c3 ) + 3.( a - b )2.c - 3.( a - b ).c2 -(a- b)3
+ c3 + 3.( a + b )2.c - 3.( a + b ).c2 - ( a + b )3
= 6.( a + b )2 .c
NA
1
30 tháng 9 2018
\(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\))
\(=a^3\left(b-c\right)-b^3\left(a-c\right)+c^3\left(a-b\right)\)
\(=a^3\left(b-c\right)-b^3\left[\left(b-c\right)+\left(a-b\right)\right]+c^3\left(a-b\right)\)
\(=a^3\left(b-c\right)-b^3\left(b-c\right)-b^3\left(a-b\right)+c^3\left(a-b\right)\)
\(=\left(b-c\right)\left(a^3-b^3\right)-\left(a-b\right)\left(b^3-c^3\right)\)
\(=\left(b-c\right)\left(a-b\right)\left(a^2+ab+b^2\right)-\left(a-b\right)\left(b-c\right)\left(b^2+bc+c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left[\left(a^2+ab+b^2\right)-\left(b^2+bc+c^2\right)\right]\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2-c^2+ab-bc\right)\)
\(=\left(a-b\right)\left(b-c\right)\left[\left(a-c\right)\left(a+c\right)+b\left(a-c\right)\right]\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(a+b+c\right)\)
NA
1
28 tháng 9 2018
\(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(=a^3\left(b-c\right)-b^3\left[a-b+b-c\right]+c^3\left(a-b\right)\)
\(=a^3\left(b-c\right)-b^3\left(a-b\right)-b^3\left(b-c\right)+c^3\left(a-b\right)\)
\(=\left(b-c\right)\left(a^3-b^3\right)-\left(a-b\right)\left(b^3-c^3\right)\)
\(=\left(b-c\right)\left(a-b\right)\left(a^2+ab+b^2\right)-\left(a-b\right)\left(b-c\right)\left(b^2+bc+c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2+ab+b^2-b^2-bc-c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2+ab-bc-c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left[\left(a-c\right)\left(a+c\right)+b\left(a-c\right)\right]\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(a+b+c\right)\)
HK
2
BH
31 tháng 5 2018
a(b-c)3+b(c-a)3+c(a-b)3
=a(b-c)3-b[(a-b)+(b-c)]+c(a-b)3
=a(b-c)3-b[(a-b)3+3(a-b)2(b-c)+3(a-b)(b-c)2+(b-c)3]
+c(a-b)3
=a(b-c)3-b(a-b)3+3b(a-b)2(b-c)+3b(a-b)(b-c)2+b(b-c)3
+c(a-b)3
=(b-c)3(a-b)-(a-b)3(b-c)-3b(a-b)(b-c)(a-b+b-c)
=(b-c)3(a-b)-(a-b)3(b-c)-3b(a-b)(b-c)(a-c)
=(a-b)(b-c)[(b-c)2-(a-b)2-3b(a-c)]
a) \(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(=a^3b-a^3c+b^3\left(c-a\right)+c^3a-c^3b\)
\(=\left(a^3b-c^3b\right)+\left(c^3a-a^3c\right)+b^3\left(c-a\right)\)
\(=-b\left(c^3-a^3\right)+ca\left(c^2-a^2\right)+b^3\left(c-a\right)\)
\(=-b\left(c-a\right)\left(c^2-ac+a^2\right)+ca\left(c+a\right)\left(c-a\right)+b^3\left(c-a\right)\)
\(=\left(c-a\right)\left(-c^2b+abc-a^2b\right)+\left(c-a\right)\left(c^2a+ca^2\right)+b^3\left(c-a\right)\)
\(=\left(c-a\right)\left(-c^2b+abc-a^2b+c^2a+ca^2+b^3\right)\)
a) a3 (b-c) + b3 (c-a) +c3 (a-b)
<=> a3b – a3c +b3c – b3a + c3a – c3b
<=> b(a3 – c3) +c(a3 – b3) + a(b3 - c3)
(Tự áp dụng hằng đẳng thức)
b)