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x6+3x4y2-8x3y3+3x2y4+y6= x6+3x4y2+3x2y4+y6-8x3y3=(x2+y2)3-(2xy)3
= (x2+y2-2xy)[(x2+y2)2+2xy(x2+y2)+(2xy)2]= (x-y)2(x4+6x2y2+y4+2x3y+2xy3)
(x2+y2-5)2-4x2y2-16xy-16=(x2+y2-5)2-(4x2y2+16xy+16)=(x2+y2-5)2-(2xy+4)2
=(x2+y2-5+2xy+4)(x2+y2-5-2xy-4)=(x2+2xy+y2-1)(x2-2xy+y2-9)=[(x+y)2-1][(x-y)2-32]=(x+y-1)(x+y+1)(x-y-3)(x-y+3)
x4+324=x4+36x2+324-36x2=(x2+18)2-(6x)2=(x2+18-6x)(x2+18+6x)
x^5+x^4+1=x^5+x^4+x^3-x^3-x^2-x+x^2+x+1=x^3(x^2+x+1)-x(x^2+x+1)+x^2+x+1=(x^3-x+1)(x^2+x+1)
\(x^5+x^4+1\)
\(=x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^3-x+1\right)\left(x^2+x+1\right)\)
Ta có : x5 - x4 + x4 - x3 - x4 + x3 - x2 + x2 - x + x - 1
= x4(x - 1) + x3(x - 1) - x3(x - 1) - x2(x - 1) + x2(x - 1) + (x - 1)
= (x4 + x3 - x3 - x2 + x2 + 1) (x - 1)
= (x4 + 1)(x - 1)
Ta có:
\(x^5+x^4+1\)
\(=x^5+x^4+x^3+1-x^3\)
\(=\left(x^5+x^4+x^3\right)+\left(1^3-x^3\right)\)
\(=x^3\left(x^2+x+1\right)+\left(1-x\right)\left(1+x+x^2\right)\)
\(=\left(x^2+x+1\right)\left(x^3+1-x\right)\)
\(a^5+a^4+1=a^5+a^4+a^3-a^3+a^2-a^2+a-a+1\)
\(=a^5+a^4+a^3-a^3-a^2-a+a^2+a+1\)
\(=\left(a^5+a^4+a^3\right)-\left(a^3+a^2+a\right)-\left(a^2+a+1\right)\)
\(=a^3\left(a^2+a+1\right)-a\left(a^2+a+1\right)+\left(a^2+a+1\right)\)
\(=\left(a^2+a+1\right)\left(a^3-a+1\right)\)
#by_Suho
\(x^8+x+1\)
\(=x^8+x^7+x^6-x^7-x^6-x^5+x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1\)
\(=x^6\left(x^2+x+1\right)-x^5\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x^2\left(x^2+x+1\right)+x^2+x+1\)
\(=\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\)
x5 + x4 + 1
= x5 + x4 + x3 - x3 + 1
= x3(x2 + x + 1) - (x3 - 1)
= x3(x2 + x + 1) - (x - 1)(x2 + x + 1)
= (x3 - x + 1)(x2 + x + 1)
1 ) \(x^5+x+1\)
\(=\left(x^5-x^2\right)+\left(x^2+x+1\right)\)
\(=x^2\left(x^3-1\right)+\left(x^2+x+1\right)\)
\(=x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^3-x^2+1\right)\)
b ) \(x^8+x^4+1\)
\(=\left(x^8+2x^4+1\right)-x^4\)
\(=\left(x^4+1\right)^2-\left(x^2\right)^2\)
\(=\left(x^4-x^2+1\right)\left(x^4+x^2+1\right)\)