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a) \(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2b-a^2c+c^2a-c^2b+b^2\left(c-a\right)\)
\(=\left(a^2b-c^2b\right)-\left(a^2c-c^2a\right)-b^2\left(a-c\right)\)
\(=b\left(a^2-c^2\right)-ac\left(a-c\right)-b^2\left(a-c\right)\)
\(=b\left(a-c\right)\left(a+c\right)-ac\left(a-c\right)-b^2\left(a-c\right)\)
\(=\left(a-c\right)\left[b\left(a+c\right)-ac-b^2\right]\)
\(=\left(a-c\right)\left(ab+bc-ac-b^2\right)\)
\(=\left(a-c\right)\left[\left(ab-b^2\right)+\left(bc-ac\right)\right]\)
\(=\left(a-c\right)\left[b\left(a-b\right)+c\left(b-a\right)\right]\)
\(=\left(a-c\right)\left[b\left(a-b\right)-c\left(a-b\right)\right]\)
\(=\left(a-c\right)\left(a-b\right)\left(b-c\right)\)
b) \(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)
\(=a^3b-a^3c+c^3a-c^3b+b^3\left(c-a\right)\)
\(=\left(a^3b-c^3b\right)-\left(a^3c-c^3a\right)-b^3\left(a-c\right)\)
\(=b\left(a^3-c^3\right)-ac\left(a^2-c^2\right)-b^3\left(a-c\right)\)
\(=b\left(a-c\right)\left(a^2+ac+c^2\right)-ac\left(a-c\right)\left(a+c\right)-b^3\left(a-c\right)\)
\(=\left(a-c\right)\left[b\left(a^2+ac+c^2\right)-ac\left(a+c\right)-b^3\right]\)
\(=\left(a-c\right)\left(ba^2+abc+bc^2-a^2c-ac^2-b^3\right)\)
\(=\left(a-c\right)\left[\left(ba^2-a^2c\right)+\left(abc-ac^2\right)+\left(bc^2-b^3\right)\right]\)
\(=\left(a-c\right)\left[a^2\left(b-c\right)+ac\left(b-c\right)+b\left(c^2-b^2\right)\right]\)
\(=\left(a-c\right)\left[a^2\left(b-c\right)+ac\left(b-c\right)-b\left(b^2-c^2\right)\right]\)
\(=\left(a-c\right)\left[a^2\left(b-c\right)+ac\left(b-c\right)-b\left(b-c\right)\left(b+c\right)\right]\)
\(=\left(a-c\right)\left(b-c\right)\left[a^2+ac-b\left(b+c\right)\right]\)
\(=\left(a-c\right)\left(b-c\right)\left(a^2+ac-b^2-bc\right)\)
\(=\left(a-c\right)\left(b-c\right)\left[\left(a-b\right)\left(a+b\right)+c\left(a-b\right)\right]\)
\(=\left(a-c\right)\left(b-c\right)\left(a-b\right)\left(a+b+c\right)\)
a: \(=ab\left(a+b\right)-bc\left(b+a\right)-bc\left(c-a\right)-ac\left(c-a\right)\)
\(=\left(a+b\right)\left(ab-bc\right)+\left(a-c\right)\left(bc-ac\right)\)
\(=\left(a+b\right)\cdot b\left(a-c\right)+\left(a-c\right)\cdot c\left(b-a\right)\)
\(=\left(a-c\right)\left(ab+b^2+cb-ac\right)\)
b: \(=ab^2+ac^2+bc^2+a^2b+a^2c+b^2c+2abc\)
\(=ab\left(a+b\right)+c^2\left(a+b\right)+c\left(a+b\right)^2\)
\(=\left(a+b\right)\left(ab+c^2+ac+cb\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(a+c\right)\)
d: \(=a^3\left(b-c\right)-b^3\left(b-c+a-b\right)+c^3\left(a-b\right)\)
\(=a^3\left(b-c\right)-b^3\left(b-c\right)-b^3\left(a-b\right)+c^3\left(a-b\right)\)
\(=\left(b-c\right)\left(a-b\right)\left(a^2+ab+b^2\right)-\left(a-b\right)\left(b-c\right)\left(b^2+bc+c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2+ab+b^2-b^2-bc-c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2+ab-bc-c^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\cdot\left[\left(a-c\right)\left(a+c\right)+b\left(a-c\right)\right]\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(a+b+c\right)\)
a) \(\left(x^2-x+2\right)^2+\left(x-2\right)^2\)
\(=\left(x^4-2x^3+5x^2-4x+4\right)+\left(x^2-4x+4\right)\)
\(=x^4-2x^3+6x^2-8x+8\)
\(=\left(x^4-2x^3+2x^2\right)+\left(4x^2-8x+8\right)\)
\(=x^2\left(x^2-2x+2\right)+4\left(x^2-2x+2\right)\)
\(=\left(x^2+4\right)\left(x^2-2x+2\right)\)
\(x^4-9x^3+28x^2-36x+16\)
\(=x^4-x^3-8x^3+8x^2+20x^2-20x-16x+16\)
\(=\left(x^4-x^3\right)-\left(8x^3-8x^2\right)+\left(20x^2-20x\right)-\left(16x-16\right)\)
\(=x^3\left(x-1\right)-8x^2\left(x-1\right)+20x\left(x-1\right)-16\left(x-1\right)\)
\(=\left(x-1\right)\left(x^3-8x^2+20x-16\right)\)
\(=\left(x-1\right)\left(x^3-2x^2-6x^2+12x+8x-16\right)\)
\(=\left(x-1\right)[x^2\left(x-2\right)-6x\left(x-2\right)+8\left(x-2\right)]\)
\(=\left(x-1\right)\left(x-2\right)\left(x^2-6x+8\right)\)
\(=\left(x-1\right)\left(x-2\right)\left(x^2-4x-2x+8\right)\)
\(=\left(x-1\right)\left(x-2\right)[x\left(x-4\right)-2\left(x-4\right)]\)
\(=\left(x-1\right)\left(x-2\right)\left(x-2\right)\left(x-4\right)\)
\(=\left(x-1\right)\left(x-2\right)^2\left(x-4\right)\)
\(a\left(b+c\right)^2\left(b-c\right)+b\left(c+a\right)^2\left(c-2\right)+c\left(a+b\right)^2\left(a-b\right)\)
\(=\left(b-c\right)\left(c-a\right)\left(c-b\right)\left(c+b+a\right)\)
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