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\(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}-\frac{3}{abc}=\left(\frac{1}{a}+\frac{1}{b}\right)^3+\left(\frac{1}{c}\right)^3-3.\frac{1}{a}.\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}\right)-\frac{3}{abc}\)
\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left[\left(\frac{1}{a}+\frac{1}{b}\right)^2-\left(\frac{1}{a}+\frac{1}{b}\right).\frac{1}{c}+\frac{1}{c^2}\right]-3.\frac{1}{a}.\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{ab}-\frac{1}{ac}-\frac{1}{bc}+\frac{1}{c^2}\right)-\frac{3}{ab}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)
\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{ab}-\frac{1}{ac}-\frac{1}{bc}\right)\)
\(\left(a^2+4b^2-5\right)^2-16\left(ab+1\right)^2\)
\(=\left(a^2+4b^2-5\right)^2-4^2\left(ab+1\right)^2\)
\(=\left(a^2+4b^2-5\right)^2-\left[4\left(ab+1\right)\right]^2\)
\(=\left(a^2+4b^2-5\right)^2-\left[4ab+4\right]^2\)
\(=\left(a^2+4b^2-5-4ab-4\right)\left(a^2+4b^2-5+4ab+4\right)\)
\(=\left(a^2+4b^2-4ab-9\right)\left(a^2+4b^2+4ab-1\right)\)
\(\left(a^2+4b^2-5\right)^2-16\left(ab+1\right)^2\)
= \(\left(a^2+4b^2-5\right)^2-\left[4\left(ab+1\right)\right]^2\)
= \(\left(a^2+4b^2-5\right)^2-\left(4ab+4\right)^2\)
= \(\left(a^2+4b^2-5-4ab-4\right)\left(a^2+4b^2-5+4ab+4\right)\)
= \(\left(a^2+4b^2-4ab-9\right)\left(a^2+4b^2+4ab-1\right)\)
= \(\left[\left(a-2b\right)^2-3^2\right]\left[\left(a+2b\right)^2-1^2\right]\)
= \(\left(a-2b-3\right)\left(a-2b+3\right)\left(a+2b-1\right)\left(a+2b+1\right)\)
x^4 + y^4=(x^2)^2+(y^2)^2
=(x^2+y^2)^2-2x^2y^2
=(x^2+y^2)^2-(√2xy)^2
=(x^2+y^2-√2 xy)(x^2+y^2+√2 xy)
\(4x^4-21x^2y^2+y^4\)
\(=\left(4x^4+4x^2y^2+y^4\right)-25x^2y^2\)
\(=\left(2x^2+y^2\right)^2-\left(5xy\right)^2\)
\(=\left(2x^2+y^2-5xy\right)\left(2x^2+y^2+5xy\right)\)
\(a,4x^4-21x^2y^2+y^4=\left(2x^2\right)^2+4x^2y^2+y^4-4x^2y^2-21x^2y^2\)
\(=\left(2x^2+y^2\right)^2-25x^2y^2\)
\(=\left(2x^2+y^2-5xy\right)\left(2x^2+y^2+5xy\right)\)
\(b,x^5-5x^3+4x=x\left(x^4-5x^2+4\right)\)
\(=x\left(x^4-4x^2-x^2+4\right)\)
\(=x\left[x^2\left(x^2-4\right)-\left(x^2-4\right)\right]\)
\(=x\left(x^2-4\right)\left(x^2-1\right)\)
\(=x\left(x-2\right)\left(x+2\right)\left(x-1\right)\left(x+1\right)\)
\(c,x^3+5x^2+3x-9=x^3-x^2+6x^2-6x+9x-9\)
\(=x^2\left(x-1\right)+6x\left(x-1\right)+9\left(x-1\right)\)
\(=\left(x-1\right)\left(x^2+6x+9\right)\)
\(=\left(x-1\right)\left(x^2+3x+3x+9\right)\)
\(=\left(x-1\right)\left[x\left(x+3\right)+3\left(x+3\right)\right]\)
\(=\left(x-1\right)\left(x+3\right)\left(x+3\right)\)
\(=\left(x-1\right)\left(x+3\right)^2\)
\(d,x^{16}+x^8-2=x^{16}+2x^8-x^8-2\)
\(=x^8\left(x^8-1\right)+2\left(x^8-1\right)\)
\(=\left(x^8-1\right)\left(x^8+2\right)\)
\(P=a^{16}+a^8b^8+b^{16}=\left(a^8\right)^2+2a^8b^8+\left(b^8\right)^2-a^8b^8\)
\(=\left(a^8+b^8\right)^2-\left(a^4b^4\right)^2\)
\(=\left(a^8+b^8+a^4b^4\right)\left(a^8+b^8-a^4b^4\right)\)
\(=\left(a^4+b^4+a^2b^2\right)\left(a^4+b^4-a^2b^2\right)\left(a^8+b^8-a^4b^4\right)\)
\(=\left(a^2+b^2+ab\right)\left(a^2+b^2-ab\right)\left(a^8+b^8-a^4b^4\right)\left(a^4+b^4-a^2b^2\right)\)