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\(x^6-9x^5+30x^4-45x^3+30x^2-9x+1\)
\(=\left(x^2\right)^3-9x^5+30x^4-45x^3+30x^2-9x+1^3\)
\(=\left(x^3-3x+1\right)^3\)
1: \(x^2-2x-24=\left(x-6\right)\left(x+4\right)\)
2: \(x^2-8x+15=\left(x-3\right)\left(x-5\right)\)
3: \(x^2-9x+14=\left(x-2\right)\left(x-7\right)\)
\(9x^2+6x-4y^2+4y=\left(9x^2+6x+1\right)-\left(4y^2-4y+1\right)=\left(3x+1\right)^2-\left(2y-1\right)^2=\left(3x+1-2y+1\right)\left(3x+1+2y-1\right)\)
a) x^2-9x+20
= x^2+4x+5x+20
=(x^2+4x)+(5x+20)
=x(x+4)+5(x+4)
=(x+4)(x+5)
\(=x^2\left(x^2+2x+1\right)+x+1\)
\(=x^2\left(x+1\right)^2+x+1\)
\(=\left(x+1\right)\left[x^2\left(x+1\right)+1\right]\)
\(=\left(x+1\right)\left(x^3+x^2+1\right)\)
\(x^4+2x^3+x^2+x+1\)
\(=x^2\left(x+1\right)^2+\left(x+1\right)\)
\(=\left(x+1\right)\left(x^3+x^2+1\right)\)
\(\left(x-5\right)\left(x-1\right)\left(x+3\right)\left(x+7\right)+60\)
\(=\left(x^2+2x-35\right)\left(x^2+2x-3\right)+60\)
\(=\left(x^2+2x\right)^2-38\left(x^2+2x\right)+105+60\)
\(=\left(x^2+2x\right)^2-3\left(x^2+2x\right)-35\left(x^2+2x\right)+165\)
\(=\left(x^2+2x-3\right)\left(x^2+2x-35\right)\)
\(=\left(x+3\right)\left(x-1\right)\left(x+7\right)\left(x-5\right)\)
x3 - 9x2 + 7x + 1
= x3 - x2 - 8x2 + 8x - x + 1
= (x3 - x2) - (8x2 - 8x) - (x - 1)
= x2(x - 1) - 8x(x - 1) - (x - 1)
= (x - 1)(x2 - 8x -1)