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a. x3+5x2+3x-9
= x3-x2+6x2-6x+9x-9
= x2(x-1)+6x(x-1)+9(x-1)
= (x2+6x+9)(x-1)
= (x+3)2(x-1)
b. x3+9x2+11x-21
= x3-x2+10x2-10x+21x-21
= x2(x-1)+10x(x-1)+21(x-1)
= (x2+10x+21)(x-1)
= (x+7)(x+3)(x-1)
c. x3-7x+6
= x3-x2+x2-x-6x+6
= x2(x-1)+x(x-1)-6(x-1)
= (x2+x-6)(x-1)
= (x+3)(x-2)(x-1)
d. x3-5x2+8x-4
= x3-x2-4x2+4x+4x-4
= x2(x-1)-4x(x-1)+4(x-1)
= (x2-4x+4)(x-1)
= (x-2)2(x-1)
e. x3-3x+2
= x3+2x2-2x2-4x+x+2
= x2(x+2)-2x(x+2)+(x+2)
= (x2-2x+1)(x+2)
= (x-1)2(x+2)
f. x3+8x2+17x+10
= x3+5x2+3x2+15x+2x+10
= x2(x+5)+3x(x+5)+2(x+5)
= (x2+3x+2)(x+5)
= (x+1)(x+2)(x+5)
g. x3+3x2+6x+4
= x3+x2+2x2+2x+4x+4
= x2(x+1)+2x(x+1)+4(x+1)
= (x2+2x+4)(x+1)
h. x3-2x-4
= x3-2x2+2x2-4x+2x-4
= x2(x-2)+2x(x-2)+2(x-2)
= (x2+2x+2)(x-2)
k. x3+x2+4
= x3+2x2-x2-2x+2x+4
= x2(x+2)-x(x+2)+2(x+2)
= (x2-x+2)(x+2)
l. x3-12x+7x-2
= x3+2x2-2x2-4x-x-2
= x2(x+2)-2x(x+2)-(x+2)
= (x2-2x-1)(x+2)
******************************************************
a) \(x^3-5x^2+8x-4=x^3-x^2-4x^2+4x+4x-4\)
\(=x^2\left(x-1\right)-4x\left(x-1\right)+4\left(x-1\right)\)
\(=\left(x-1\right)\left(x^2-4x+4\right)=\left(x-1\right)\left(x-2\right)^2\)
b) \(x^3-3x+2=x^3+2x^2-2x^2-4x+x+2\)
\(=x^2\left(x+2\right)-2x\left(x+2\right)+\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x+1\right)=\left(x+2\right)\left(x-1\right)^2\)
c) \(x^3-5x^2+3x+9=x^3+x^2-6x^2-6x+9x+9\)
\(=x^2\left(x+1\right)-6x\left(x+1\right)+9\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-6x+9\right)=\left(x+1\right)\left(x-3\right)^2\)
d) \(x^3+8x^2+17x+10=x^3+2x^2+6x^2+12x+5x+10\)
\(=x^2\left(x+2\right)+6x\left(x+2\right)+5\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2+6x+5\right)=\left(x+2\right)\left(x+5\right)\left(x+1\right)\)
e) \(x^3+3x^2+6x+4=x^3+x^2+2x^2+2x+4x+4\)
\(=x^2\left(x+1\right)+2x\left(x+1\right)+4\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2+2x+4\right)\)
f) \(x^3+3x^2+3x+2=x^3+2x^2+x^2+2x+x+2\)
\(=x^2\left(x+2\right)+x\left(x+2\right)+\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2+x+1\right)\)