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a)x^2-(a+b)x+ab
= x^2 - ax - bx + ab
= (x^2 - ax) - (bx - ab)
= x(x-a) - b(x-a)
= (x-b)(x-a)
b)7x^3-3xyz-21x^2+9z
=
c)4x+4y-x^2(x+y)
= 4(x + y) - x^2(x+y)
= (4-x^2) (x+y)
= (2-x)(2+x)(x+y)
d) y^2+y-x^2+x
= (y^2 - x^2) + (x+y)
= (y-x)(y+x)+ (x+y)
= (y-x+1) (x+y)
e)4x^2-2x-y^2-y
= [(2x)^2 - y^2] - (2x +y)
= (2x-y)(2x+y) - (2x+y)
= (2x -y -1)(2x+y)
f)9x^2-25y^2-6x+10y
=
a) 1/2(x3+8)=1/2(x+2)(x2-2x+4)
b) x4(x-y)+2x3(x-y)=x3(x+2)(x-y)
c) x2-(y2-6y+9)=x2-(y-3)2=(x-y+3)(x+y-3)
d) xy(x3+y3)=xy(x+y)(x2-xy+y2)
e)3x2(x2-25y2)=3x2(x-5y)(x+5y)
f) 4x4+4x2y2+y4-4x2y2= (2x2+y2)2-(2xy)2=(2x2-2xy+y2)(2x2+2xy+y2)
a) \(\frac{1}{2}x^3+4=\frac{1}{2}\left(x^3+8\right)=\frac{1}{2}\left(x+2\right)\left(x^2-2x+4\right)\)
b) \(x^5-x^4y+2x^4-2x^3y=x^3\left(x^2-xy+2x-2y\right)=x^3\left[x\left(x-y\right)+2\left(x-y\right)\right]=x^2\left(x-y\right)\left(x+2\right)\)
c) \(x^2-y^2+6y-9=x^2-\left(y-3\right)^2=\left(x+y-3\right)\left(x-y+3\right)\)
d) \(x^4y+xy^4=xy\left(x^3+y^3\right)=xy\left(x+y\right)\left(x^2-xy+y^2\right)\)
e) \(3x^4-75x^2y^2=3x^2\left(x^2-25y^2\right)=3x^2\left(x+5y\right)\left(x-5y\right)\).
f) \(4x^4+y^4=\left(2x^2+y^2\right)^2-\left(2xy\right)^2=\left(2x^2+y^2+2xy\right)\left(2x^2-y^2-2xy\right)\)
a) \(14\left(x-y\right)^2+21\left(y-x\right)\)
\(=14\left(x-y\right)^2-21\left(x-y\right)\)
\(=7\left(x-y\right)\left[2\left(x-y\right)-3\right]\)
\(=7\left(x-y\right)\left(2x-2y-3\right)\)
b) \(7x^5\left(y-3\right)-49x^4\left(3-y\right)^3\)
\(=7x^4\left(y-3\right)\left[x+7\left(y-3\right)^2\right]\)
\(=7x^4\left(y-3\right)\left(x+7y^2-42y+63\right)\)
c) \(\left(x^2-9\right)^2-x^2\left(x-3\right)^2\)
\(=\left(x-3\right)^2\left(x+3\right)^2-x^2\left(x-3\right)^2\)
\(=\left(x-3\right)^2\left[\left(x+3\right)^2-x^2\right]\)
\(=\left(x-3\right)^2\left(x^2+6x+9-x^2\right)\)
\(=3\left(x-3\right)^2\left(x+3\right)\)
d) \(\left(4x^2-1\right)^2-9\left(2x-1\right)^2\)
\(=\left(2x-1\right)^2\left(2x+1\right)^2-9\left(2x-1\right)^2\)
\(=\left(2x-1\right)^2\left[\left(2x+1\right)^2-9\right]\)
\(=\left(2x-1\right)^2\left(4x^2+4x+1-9\right)\)
\(=4\left(2x-1\right)^2\left(x^2+x-2\right)\)
\(=4\left(2x-1\right)^2\left(x-1\right)\left(x+2\right)\)