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\(a,6x^3-9x^2=3x^2\left(2x-3\right)\)
\(b,4x^2y-8xy^2+10x^2y^2=2xy\left(2x-4y+5xy\right)\)
\(c,20x^2y-12x^3=4x^2\left(5y-3x\right)\)
\(d,4xy^2+8xyz=4xy\left(y+2z\right)\)
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)
a.\(3x^2-11x+6\)
= \(3x^2-9x-2x+6\)
=\(3x\left(x-3\right)-2\left(x-3\right)\)
=\(\left(x-3\right)\left(3x-2\right)\)
b\(8x^2+10x-3\)
=.\(8x^2-2x+12x-3\)
=\(2x\left(4x-1\right)+3\left(4x-1\right)\)
=\(\left(4x-1\right)\left(2x+3\right)\)
d.\(x^2-y^2+10x-6y+16\)
=\(\left(x^2+10x+25\right)-\left(y^2+6y+9\right)\)
=\(\left(x+5\right)^2-\left(y+3\right)^2\)
=\(\left(x+5-y-3\right)\left(x+5+y+3\right)\)
=\(\left(x-y+2\right)\left(x+y+8\right)\)
e.\(x^4+x^2y^2+y^4\)
=\(x^4+2x^2y^2+y^4-x^2+y^2\)
=\(\left(x^2+y^2\right)^2-x^2y^2\)
=\(\left(x^2+y^2-xy\right)\left(x^2+y^2+xy\right)\)
a)
\(=3x^2-9x-2x+6=3x\left(x-3\right)-2\left(x-3\right)=\left(x-3\right)\left(3x-2\right)\)
a) \(6x^3-12x^2y^2+6xy^3=6x.\left(x^2-2xy^2+y^3\right)\)
b) \(\left(x^2+4\right)^2-16=\left(x^2+4-4\right)\left(x^2+4+4\right)=x^2\left(x^2+8\right)\)
c) \(5x^2-5xy-10x+10y=\left(5x^2-5xy\right)-\left(10x-10y\right)=5x\left(x-y\right)-10\left(x-y\right)\)
\(=\left(x-y\right)\left(5x-10\right)=5\left(x-y\right)\left(x-2\right)\)
d) \(a^3-3a+3b-b^3=\left(a^3-b^3\right)-\left(3a-3b\right)=\left(a-b\right)\left(a^2+ab+b^2\right)-3.\left(a-b\right)\)
\(=\left(a-b\right)\left(x^2+ab+b^2-3\right)\)
e) \(x^2-2x-y^2+1=\left(x^2-2x+1\right)-y^2=\left(x-1\right)^2-y^2=\left(x-1-y\right)\left(x-1+y\right)\)
f) \(x^2-x-2=x^2+x-2x-2=\left(x^2+x\right)-\left(2x+2\right)=x\left(x+1\right)-2\left(x+1\right)\)
\(=\left(x+1\right)\left(x-2\right)\)
g) \(x^4-5x^2+4=x^4-4x^2+4-x^2=\left(x^4-4x^2+4\right)-x^2=\left(x^2-2\right)^2-x^2\)
\(=\left(x^2-2-x\right)\left(x^2-2+x\right)\)
j) \(x^3-x^3-2x^2-x=-2x^2-x=-\left(2x^2+x\right)=-x\left(2x+1\right)\)
k) \(\left(a^3-27\right)-\left(3-a\right)\left(6a+9\right)=\left(a-3\right).\left(a^2+3a+9\right)+\left(a-3\right)\left(6a+9\right)\)
\(\left(a-3\right)\left(a^2+3a+9+6a+9\right)=\left(a-3\right)\left(a^2+9a+18\right)\)
h) \(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z+y^2z-y^2x+z^2x-z^2y\)
\(=\left(x^2y-y^2x\right)-\left(x^2z-y^2z\right)+\left(z^2x-z^2y\right)\)
\(=xy\left(x-y\right)-z\left(x^2-y^2\right)+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left(xy-zx-zy+z^2\right)\)
\(=\left(x-y\right)\left[\left(xy-zx\right)-\left(zy-z^2\right)\right]\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)
\(\left(x-y\right)\left(y-z\right)\left(x-z\right)\)
a: \(6x^2-3x\)
\(=3x\cdot2x-3x\)
=3x(2x-1)
b: \(15x^5y^4+10x^3y^3-5xy\)
\(=5xy\cdot3x^4y^3+5xy\cdot2x^2y^2-5xy\cdot1\)
\(=5xy\left(3x^4y^3+2x^2y^2-1\right)\)
c: \(x^2y+4xy+4y\)
\(=y\cdot x^2+y\cdot4x+y\cdot4\)
\(=y\left(x^2+4x+4\right)=y\left(x+2\right)^2\)