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\(3x^3-7x^2+17x-5=3x^3-x^2-6x^2+2x+15x-5\)
\(=x^2\left(3x-1\right)-2x\left(3x-1\right)+5\left(3x-1\right)\)
\(=\left(3x-1\right)\left(x^2-2x+5\right)\)
\(x^3-x^2-4=x^3+x^2+2x-2x^2-2x-4\)
\(=x\left(x^2+x+2\right)=2\left(x^2+x+2\right)=\left(x-2\right)\left(x^2+x+2\right)\)
x2 - 10xy + 9y
= x2 - xy - 9xy + 9y2
= x(x - y) - 9y(x - y)
= (x - y)(x - 9y)
x3 - x2 - 4
= x3 + x2 + 2x - 2x2 - 2x - 4
= x(x2 + x + 2) - 2(x2 + x + 2)
= (x2 + x + 2)(x - 2)
x3 - 5x2 + 8x - 4
= x3 - x2 - 4x2 + 4x + 4x - 4
= x2(x - 1) - 4x(x - 1) + 4(x - 1)
= (x - 1)(x2 - 4x + 4)
= (x - 1)(x - 2)2
x3 + 2x - 3
= x3 - x2 + x2 - x + 3x - 3
= x2(x - 1) + x(x - 1) + 3(x - 1)
= (x - 1)(x2 + x + 3)
x3 + 5x2 + 8x + 4
= x3 + x2 + 4x2 + 4x + 4x + 4
= x2(x + 1) + 4x(x + 1) + 4(x + 1)
= (x + 1)(x2 + 4x + 4)
= (x + 1)(x + 2)2
a, \(=x^3-4x+x^2+4x+4=x\left(x-2\right)\left(x+2\right)+\left(x+2\right)^2\)
\(=\left(x+2\right)\left[x\left(x-2\right)+x+2\right]=\left(x+2\right)\left(x^2-x+2\right)\)
b, \(=2x^3-x^2-2x^2+3x-1=x^2\left(2x-1\right)-\left(x-1\right)\left(2x-1\right)\)
\(=\left(2x-1\right)\left(x^2-x+1\right)\)
c, \(=x^4+x^3+x^3+x^2+x+1=x^3\left(x+1\right)+x^2\left(x+1\right)+x+1\)
\(=\left(x+1\right)\left(x^3+x^2+1\right)\)
4x3 - 13x2 + 9x - 18
= 4x3 - 12x2 - x2 + 3x + 6x - 18
= 4x2(x - 3) - x(x - 3) + 6(x - 3)
= (x - 3)(4x2 - x + 6)
x2 + 5x - 6
= x2 + 2x + 3x - 6
= x(x + 2) - 3(x + 2)
= (x + 2)(x - 3)
x3 + 8x2 + 17x + 10
= x3 + x2 + 7x2 + 7x + 10x + 10
= x2(x + 1) + 7x(x + 1) + 10(x + 1)
= (x + 1)(x2 + 7x + 10)
= (x + 1)(x2 + 5x + 2x + 10)
= (x + 1)[ x(x + 5) + 2(x + 5)]
= (x + 1)(x + 5)(x + 2)
x3 + 3x2 + 6x + 4
= x3 + 3x2 + 3x + 1 + 3x + 3
= (x + 1)3 + 3(x + 1)
= (x + 1)[(x + 1)2 + 3]
= (x + 1)(x2 + 2x + 1 + 3)
= (x + 1)(x2 + 2x + 4)
2x3 - 12x2 + 17x - 2
= 2x3 - 8x2 - 4x2 + x + 16x - 2
= (2x3 - 8x2 + x) - (4x2 - 16x + 2)
= x(2x2 - 8x + 1) - 2(2x2 - 8x + 1)
= (2x2 - 8x + 1)(x - 2)
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