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a) \(-4x^2+12xy-9y^2+25=-\left(2x\right)^2+2.2x.3y-\left(3y\right)^2+25\)
= \(-\left(2x-3y\right)^2+25=\left(5-2x+3y\right)\left(5+2x-3y\right)\)
b) \(x^3-3x^2+3x-1=\left(x^3-1\right)-\left(3x^2-3x\right)=\left(x-1\right)\left(x^2+x+1^2\right)-3x\left(x-1\right)\)
= \(\left(x-1\right)\left(x^2+x+1^2-3x\right)=\left(x-1\right)\left(x^2-2x+1\right)=\left(x-1\right)\left(x-1\right)^2=\left(x-1\right)^3\)
c) \(a^6-a^4+2a^3+2a^2=a^2\left(a^4-a^2+2a+2\right)=a^2\left[a^2\left(a^2-1\right)+2\left(a+1\right)\right]\)
= \(a^2\left[a^2\left(a+1\right)\left(a-1\right)+2\left(a+1\right)\right]=a^2\left(a+1\right)\left(a^3+a^2+2\right)\)
a)\(\dfrac{27-x^3}{5x+5}:\dfrac{2x-6}{3x+3}\)
\(=\dfrac{\left(3-x\right)\left(9+3x+x^2\right)}{5\left(x+1\right)}:\dfrac{2\left(x-3\right)}{3\left(x+1\right)}\)
\(=\dfrac{\left(3-x\right)\left(9+3x+x^2\right)3\left(x+1\right)}{5\left(x+1\right)2\left(x-3\right)}\)
\(=\dfrac{-\left(x-3\right)\left(9+3x+x^2\right)3\left(x+1\right)}{5\left(x+1\right)2\left(x-3\right)}\)
\(=\dfrac{-\left(9+3x+x^2\right)3}{10}\)
b)\(4x^2-16:\dfrac{3x+6}{7x-2}\)
\(=4\left(x^2-4\right):\dfrac{3\left(x+2\right)}{7x-2}\)
\(=4\left(x-2\right)\left(x+2\right)\cdot\dfrac{7x-2}{3\left(x+2\right)}\)
\(=\dfrac{4\left(x-2\right)\left(x+2\right)\left(7x-2\right)}{3\left(x+2\right)}\)
\(=\dfrac{4\left(x-2\right)\left(7x-2\right)}{3}\)
c)\(\dfrac{3x^3+3}{x-1}:x^2-x+1\)
\(=\dfrac{3\left(x^3+1\right)}{x-1}:x^2-x+1\)
\(=\dfrac{3\left(x+1\right)\left(x^2-x+1\right)}{x-1}\cdot\dfrac{1}{x^2-x+1}\)
\(=\dfrac{3\left(x+1\right)}{x-1}\)
d)\(\dfrac{4x+6y}{x-1}:\dfrac{4x^2+12xy+9y^2}{1-x^3}\)
\(=\dfrac{2\left(2x+3y\right)}{x-1}\cdot\dfrac{\left(1-x\right)\left(1+x+x^2\right)}{\left(2x+3y\right)^2}\)
\(=\dfrac{2\left(2x+3y\right)}{x-1}\cdot\dfrac{-\left(x-1\right)\left(1+x+x^2\right)}{\left(2x+3y\right)^2}\)
\(=\dfrac{-2\left(1+x+x^2\right)}{2x+3y}\)
a) \(\dfrac{27-x^3}{5x+5}:\dfrac{2x-6}{3x+3}\)
\(=\dfrac{27-x^3}{5x+5}.\dfrac{3x+3}{2x-6}\)
\(=\dfrac{\left(3-x\right)\left(9+3x+x^2\right)}{5\left(x+1\right)}.\dfrac{3\left(x+1\right)}{2\left(x-3\right)}\)
\(=-\dfrac{3\left(x-3\right)\left(x^2+3x+9\right)\left(x+1\right)}{10\left(x+1\right)\left(x-3\right)}\)
\(=-\dfrac{3\left(x^2+3x+9\right)}{10}\)
b) \(4x^2-16:\dfrac{3x+6}{7x-2}\)
\(=4x^2-16.\dfrac{7x-2}{3x+6}\)
\(=\dfrac{4\left(x^2-4\right)\left(7x-2\right)}{3\left(x+2\right)}\)
\(=\dfrac{4\left(x-2\right)\left(x+2\right)\left(7x-2\right)}{3\left(x+2\right)}\)
\(=\dfrac{4\left(x-2\right)\left(7x-2\right)}{3}\)
c) \(\dfrac{3x^3+3}{x-1}:x^2-x+1\)
\(=\dfrac{3x^3+3}{x-1}.\dfrac{1}{x^2-x+1}\)
\(=\dfrac{3\left(x^3+1\right)}{\left(x-1\right)\left(x^2-x+1\right)}\)
\(=\dfrac{3\left(x+1\right)\left(x^2-x+1\right)}{\left(x-1\right)\left(x^2-x+1\right)}\)
\(=\dfrac{3\left(x+1\right)}{x-1}\)
d) \(\dfrac{4x+6y}{x-1}:\dfrac{4x^2+12xy+9y^2}{1-x^3}\)
\(=\dfrac{4x+6y}{x-1}.\dfrac{1-x^3}{4x^2+12xy+9y^2}\)
\(=\dfrac{2\left(2x+3y\right)\left(1-x\right)\left(1+x+x^2\right)}{\left(x-1\right)\left(2x+3y\right)^2}\)
\(=-\dfrac{2\left(2x+3y\right)\left(x-1\right)\left(x^2+x+1\right)}{\left(x-1\right)\left(2x+3y\right)^2}\)
\(=-\dfrac{2\left(x^2+x+1\right)}{2x+3y}\)
a) \(4x^2-1=\left(2x\right)^2-1^2=\left(2x-1\right)\left(2x+1\right)\)
b) \(25a^2-0.01=\left(5a\right)^2-\left(0.1\right)^2=\left(5a-0.1\right)\left(5a+0.1\right)\)
a, \(4x^2-1\)
=\(\left(2x^2\right)-1\)
=\(\left(2x-1\right).\left(2x+1\right)\)
b,\(25a^2-0,01\)
=\(\left(5a\right)^2-\left(0,1\right)^2\)
=\(\left(5a-0,1\right)\left(5a+0,1\right)\)
c,2a(x+y)-3b(x+y)
=(x+y)(2a-3b)
d,\(4x^2-12xy+9y^2\)
=\(\left(2x-3y\right)^2\)
g,\(\left(x+y\right)^2-2\left(x+y\right)+1\)
=\(\left(x+y-1\right)^2\)
h,\(x^4-9x^3+x^2-9x\)
=\(x^3\left(x-9\right)+x\left(x-9\right)\)
=\(\left(x^3+x\right).\left(x-9\right)\)
=\(x.\left(x^2+1\right).\left(x-9\right)\)
i,\(x^2-x-y^2-y\)
=\(x^2-y^2-x-y\)
=\(\left(x-y\right).\left(x+y\right)-\left(x+y\right)\)
=\(\left(x+y\right).\left(x-y-1\right)\)
a)x4+2x2+1=(x2)2+2.x2.1+12=(x2+1)2
b) 4x2-12xy+9y2=(2x)2-2.2x.3y+(3y)2=(2x-3y)2
c) -x2-2xy-y2= -(x2+2xy+y2)=-(x+y)2
d) (x+y)2-2(x+y)+1= (x+y-1)2
e)x3-3x2+3x-1= (x3-1)+(-3x2+3x)=(x-1)(x2+x+1)-3x(x-1)
=(x-1)(x2-2x+1)=(x-1)3
g) x3+6x2+12x+8= (x+2)3
h) x3+1-x2-x= (x+1)(x2-x+1)-(x2+x)=(x+1)(x2-x+1)-x(x+1)
=(x+1)(x-1)2
k) (x+y)3-x3-y3= (x+y)3-(x3+y3)=(x+y)3-(x+y)(x2-xy+y2)
= (x+y)[(x+y)2-(x2-xy+y2)]
= 3xy(x+y)
\(3y^2\left(a-3x\right)-a\left(a-3x\right)=\left(3y^2-a\right)\left(a-3x\right)\)
\(2a^2-3b-6a+ab=a\left(2a+b\right)-3\left(2a+b\right)=\left(a-3\right)\left(2a+b\right)\)
\(\left(a-b\right)^2-4=\left(a-b\right)^2-2^2=\left(a-b-2\right)\left(a-b+2\right)\)
\(4x^2-12xy+3x-9y=4x\left(x-3y\right)+3\left(x-3y\right)=\left(x-3y\right)\left(4x+3\right)\)