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a) \(x^2-x+\frac{1}{4}=\left(x-\frac{1}{2}\right)^2\)
b) \(49x^2+28x+4=\left(7x+2\right)^2\)
c) \(4x^2-25y^2=\left(2x-5y\right)\left(2x+5y\right)\)
d) \(x^3-8x^2y+27xy^2-27y^3=\left(x-3y\right)^3\)
e) \(8+125x^3=\left(2+5x\right)\left(4-10x+25x^2\right)\)
f) \(27x^3-64=\left(3x-4\right)\left(9x^2+12x+16\right)\)
a/ \(x^2-x+\frac{1}{4}\) \(=x^2-2.\frac{1}{2}x+\left(\frac{1}{2}\right)^2\) \(=\left(x-\frac{1}{2}\right)^2\)
b/ \(49x^2+28x+4\) \(=\) \(\left(7x\right)^2+2.7x.2+2^2\) \(=\left(7x+2\right)^2\)
c/ \(4x^2-25y^2\) \(=\left(2x\right)^2-\left(5y\right)^2\) \(=\left(4x-5y\right)\left(4x+5y\right)\)
d/\(x^3-9x^2y+27xy^2-27y^3\) \(=x^3-3.x^2.3y+3.x.9y^2-\left(3y\right)^3\) \(=\left(x-3y\right)^3\)
e/ \(8+125x^3\)\(=\left(2+5x\right)\left(4-10x+25x\right)\)
d/ \(27x^3-64=\left(3x-4\right)\left(9x^2+12x+16\right)\)
1.a) (4x - 6y)2 - (8xy - 5)2 = (4x - 6y - 8xy + 5)(4x - 6y + 8xy - 5)
b) 16x2 - 49y2 = (4x)2 - (7y)2 = (4x - 7y)(4x + 7y)
c) 36x2 + 60x + 25 = (6x)2 + 2.6x.5 + 52 = (6x + 5)2
d) (2x - y)(x - y) - (3y - 4x)2 + (y - 2x)(2y - 3x) = (y - 2x)(y - x) + (y - 2x)(2y - 3x) - (3y - 4x)2
= (y - 2x)[(y - x) + (2y - 3x)] - (3y - 4x)2 = (y - 2x)(3y - 4x) - (3y - 4x)2 = (3y - 4x)[(y - 2x) - (3y - 4x)] = 2(3y - 4x)(x - y)
2.M = (3x - 4)(9x2 - 12x + 16) + (6x - 8)2 = (3x - 4)[(3x)2 - 2.3x.4 + 42] + [2(3x - 4)]2 = (3x - 4)(3x - 4)2 + 4(3x - 4)2
= (3x - 4)2(3x - 4 + 4) = 3x(3x - 4)2
b) \(-4x^2-4x-1\)
\(=-\left(4x^2+4x+1\right)\)
\(=-\left(2x+1\right)^2\)
c) \(\frac{4}{9}x^2-25y^2\)
\(=\left(\frac{2}{3}x+5y\right)\left(\frac{2}{3}x-5y\right)\)
d) \(\frac{1}{27}x^3-8\)
\(=\left(\frac{1}{3}x-2\right)\left(\frac{1}{9}x+\frac{2}{3}x+4\right)\)
a) \(-x^3+9x^2-27x+27=-\left(x^3-3.3.x^2+3.3^2.x-3^3\right)=-\left(x-3\right)^3\)
b)\(x^4-2x^3-x^2+2x+1=x^4+\left(-x\right)^2+\left(-1\right)^2+2x^2\left(-x\right)+2.\left(-x\right).\left(-1\right)+2x^2.\left(-1\right)\)
\(=\left(x^2-x-1\right)^2\)
c)\(8x^3+27y^3+36x^2y+54xy^2=\left(2x\right)^3+3.\left(2x\right)^2.3y+3.2x.\left(3y\right)^2+\left(3y\right)^3\)
\(=\left(2x+3y\right)^2\)