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\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
a) Biến đổi VT ta có :
(a2-b2)2 + (2ab)2
= a4 -2a2+b4+4a2b2
= a4+2a2b2 +b4
= (a2b2)2 = VP (đpcm)
b) Biến đổi vế trái ta có :
(ax+b)2 + (a-bx)2+cx2+c2
= a2x2+2axb+b2 +a2 - 2axb+b2x2 +c2x2+ c2
= (a2+b2+c2) + x2(a2+b2+c2)
= (a2+b2+c2) (x2+1) = VP (đpcm)
b: \(=ab^2+ac^2+abc+bc^2+ba^2+abc+a^2c+b^2c+abc\)
\(=ab\left(a+b+c\right)+bc\left(a+b+c\right)+ac\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(ab+bc+ac\right)\)
a: \(=\left(x^2-x^2y^2\right)+\left(y^2-y\right)+\left(xy-x\right)\)
\(=-x^2\left(y-1\right)\left(y+1\right)+y\left(y-1\right)+x\left(y-1\right)\)
\(=\left(y-1\right)\left(-x^2y-x^2+y+x\right)\)
\(=\left(1-y\right)\left(x^2y+x^2-x-y\right)\)
\(=\left(1-y\right)\cdot\left[y\left(x-1\right)\left(x+1\right)+x\left(x-1\right)\right]\)
\(=\left(1-y\right)\left(x-1\right)\left(xy+y+x\right)\)
a) \(bc\left(b+c\right)+ca\left(c-a\right)-ab\left(a+b\right)\)
\(=bc\left(b+c\right)+ca\left(c-a\right)-ab\left[\left(b+c\right)-\left(c-a\right)\right]\)
\(=bc\left(b+c\right)+ca\left(c-a\right)-ab\left(b+c\right)+ab\left(c-a\right)\)
\(=\left(b+c\right)\left(bc-ab\right)+\left(c-a\right)\left(ca+ab\right)\)
\(=\left(b+c\right)\left(c-a\right)b+\left(c-a\right)\left(b+c\right)a\)
\(=\left(b+a\right)\left(c-a\right)\left(c+b\right)\)
b) \(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left[\left(b-c\right)+\left(c-a\right)\right]\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left(b-c\right)-c^2\left(c-a\right)\)
\(=\left(b-c\right)\left(a^2-c^2\right)+\left(c-a\right)\left(b^2-c^2\right)\)
\(=\left(b-c\right)\left(a-c\right)\left(a+c\right)+\left(c-a\right)\left(b-c\right)\left(b+c\right)\)
\(=\left(b-c\right)\left(c-a\right)\left(b+c-a-c\right)\)
\(=\left(b-a\right)\left(b-c\right)\left(c-a\right)\)
a) \(a\left(b^2+c^2+bc\right)+b\left(c^2+a^2+ac\right)+c\left(a^2+b^2+ab\right)\)
\(=ab^2+ac^2+abc+bc^2+ba^2+abc+ca^2+cb^2+abc\)
\(=\left(ab^2+abc+ba^2\right)+\left(ac^2+ca^2+abc\right)+\left(bc^2+abc+cb^2\right)\)
\(=ab\left(b+c+a\right)+ac\left(c+a+b\right)+bc\left(c+a+b\right)\)
\(=\left(a+b+c\right)\left(ab+ac+bc\right)\)
b) \(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=ab^2+ac^2+abc+bc^2+ba^2+abc+ca^2+cb^2+abc-abc\)
\(=\left(ab^2+ba^2\right)+\left(ac^2+bc^2\right)+\left(abc+cb^2\right)+\left(abc+ca^2\right)\)
\(=ab\left(a+b\right)+c^2\left(a+b\right)+cb\left(a+b\right)+ca\left(b+a\right)\)
\(=\left(a+b\right)\left(ab+c^2+bc+ac\right)\)
\(=\left(a+b\right)\left[a\left(b+c\right)+c\left(c+b\right)\right]\)
\(=\left(a+b\right)\left(b+c\right)\left(a+c\right)\)
c) \(a\left(a+2b\right)^3-b\left(2a+b\right)^3\)
\(=a\left(a^3+3a^2.2b+3a4b^2+8b^3\right)-b\left(8a^3+3.4a^2.b+3.2a.b^2+b^3\right)\)
\(=a\left(a^3+6a^2b+12ab^2+8b^3\right)-b\left(8a^3+12a^2b+6ab^2+b^3\right)\)
\(=a^4+6a^3b+12a^2b^2+8b^3a-8a^3b-12a^2b^2-6ab^3-b^4\)
\(=a^4+6a^3b+8b^3a-8a^3b-6ab^3-b^4\)
\(=\left(a^4-b^4\right)+\left(6a^3b-6ab^3\right)+\left(8b^3a-8a^3b\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2\right)+6ab\left(a^2-b^2\right)+8ab\left(b^2-a^2\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2\right)+6ab\left(a^2-b^2\right)-8ab\left(a^2-b^2\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2+6ab-8ab\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a^2+b^2-2ab\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a-b\right)^2\)
\(=\left(a-b\right)^3\left(a+b\right)\)
ôi cảm ơn b nhiều lắm