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a) \(x^3-x^2-4=x^3-2x^2+x^2-4=x^2\left(x-2\right)+\left(x-2\right)\left(x+2\right)=\left(x-2\right)\left(x^2+x+2\right)\)
b) \(x^3-5x^2+8x-4=x^3-x^2-4x^2+4x+4x-4=x^2\left(x-1\right)-4x\left(x-1\right)+4\left(x-1\right)\)
\(=\left(x-1\right)\left(x^2-4x+4\right)=\left(x-1\right)\left(x-2\right)^2\)
c) \(2x^3-12x^2+17x-2=2x^3-4x^2-8x^2+16x+x-2=2x^2\left(x-2\right)-8x\left(x-2\right)+\left(x-2\right).\)
\(=\left(x-2\right)\left(2x^2-8x+1\right)\)
d) \(2x^4+x^3-22x^2+15x+36=2x^4+2x^3-x^3-x^2-21x^2-21x+36x+36.\)
\(=2x^3\left(x+1\right)-x^2\left(x+1\right)-21x\left(x+1\right)+36\left(x+1\right)\)
\(=\left(x+1\right)\left(2x^3-x^2-21x+36\right)\)
a) x2 + 6x + 9 = x2 + 2 . x . 3 + 32 = (x + 3)2
b) 10x – 25 – x2 = -(-10x + 25 +x2) = -(25 – 10x + x2)
= -(52 – 2 . 5 . x – x2) = -(5 – x)2
c) 8x3 - 1/8 = (2x)3 – (1/2)3 = (2x - 1/2)[(2x)2 + 2x . 12 + (1/2)2]
= (2x - 1/2)(4x2 + x + 1/4)
d)1/25x2 – 64y2 = (1/5x)2(1/5x)2- (8y)2 = (1/5x + 8y)(1/5x - 8y)
\(3x^2-5x+2=3x^2-3x-2x+2=3x\left(x-1\right)-2\left(x-1\right)=\left(3x-2\right)\left(x-1\right)\)
b.)x^4+5x^3+15x-9
=x^4-9+5x^3+15x
=(x^2-3)(x^2+3)+5x(x^2+3)
=(x^2+3)(x^2-3+5x)
1,\(=x^3-2x^2+x^2-2x-6x+12=x^2\left(x-2\right)+x\left(x-2\right)-6\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2-2x+3x-6\right)=\left(x-2\right)^2\left(x+3\right)\)
2,\(=x^3-5x^2-4x^2+20x-5x+25=x^2\left(x-5\right)-4x\left(x-5\right)-5\left(x-5\right)\)
\(=\left(x-5\right)\left(x^2+x-5x-5\right)=\left(x-5\right)^2\left(x+1\right)\)
a) \(12x^3+8x^2-3x-2=4x^2\left(3x+2\right)-\left(3x+2\right)\)
\(=\left(3x+2\right)\left(4x^2-1\right)=\left(3x+2\right)\left(2x-1\right)\left(2x+1\right)\)
b) \(18x^3+27x^2-2x-3=9x^2\left(2x+3\right)-\left(2x+3\right)\)
\(=\left(2x+3\right)\left(9x^2-1\right)=\left(2x+3\right)\left(3x-1\right)\left(3x+1\right)\)
c) \(8x^3+4x^2-34x+15=4x^2\left(2x-3\right)+8x\left(2x-3\right)-5\left(2x-3\right)\)
\(=\left(2x-3\right)\left(4x^2+8x-5\right)=\left(2x-3\right)\left(2x-1\right)\left(2x+5\right)\)
a) 4x2 - 20x + 25 - 36y2
= (2x - 5)2 - 36y2
= (2x - 5 - 6y)(2x - 5 + 6y)
b) x3 + x2 - 2x - 8
= (x3 - 8) + (x2 - 2x)
= (x - 2)(x2 + 2x + 4) + x(x - 2)
= (x - 2)(x2 + 2x + 4 + x)
= (x - 2)(x2 + 3x + 4)
d) x4 + 6x3 + 9x2 - 16
= x2(x2 + 6x + 9) - 16
= x2(x + 3)2 - 16
= (x2 + 3x)2 - 16
= (x2 + 3x - 4)(x2 + 3x + 4)
= (x2 + 4x - x - 4)(x2 + 3x + 4)
= [x(x + 4) - (x + 4)](x2 + 3x + 4)
= (x - 1)(x + 4)(x2 + 3x + 4)
a) \(6x^2-x-1\)
\(=6x^2-3x+2x-1\)
\(=3x\left(2x-1\right)+\left(2x-1\right)\)
\(=\left(3x+1\right)\left(2x-1\right)\)
\(x^3+x^2+9x-10x^2-10x+25x+25\)
\(=x^2\left(x+1\right)-10x\left(x+1\right)+25\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-10x+25\right)=\left(x+1\right)\left(x-5\right)^2\)
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