Question

Answer 1

a) \(z^2-\left(x-1\right)^2+2\left(x-1\right)-1\)

\(=z^2-\left[\left(x-1\right)^2-2\left(x-1\right)+1\right]\\ =z^2-\left(x-1-1\right)^2\\ =z^2-\left(x-2\right)^2\\ =\left(z-x+2\right)\left(z+x-2\right)\)

b)

 \(\left(2x-1\right)^3-4x^2\left(2x-1\right)\\ =\left(2x-1\right)\left[\left(2x-1\right)^2-4x^2\right]\\ =\left(2x-1\right)\left(2x-1-2x\right)\left(2x-1+2x\right)\\ =-\left(2x-1\right)\left(4x-1\right)\)

c)

 \(x^2-6x+8\\ =\left(x^2-6x+9\right)-1\\ =\left(x-3\right)^2-1^2\\ =\left(x-3-1\right)\left(x-3+1\right)\\ =\left(x-4\right)\left(x-2\right)\)

Answer 2

a) \(z^2-\left(x-1\right)^2+2\left(x-1\right)-1=z^2-\left(x-1\right)\left[\left(x-1\right)+2\right]-1\)

Answer 3

a: \(z^2-\left(x-1\right)^2+2\left(x-1\right)-1\)

\(=z^2-\left(x-1-1\right)^2\)

\(=z^2-\left(x-2\right)^2\)

\(=\left(z-x+2\right)\left(x+z-2\right)\)

b: \(\left(2x-1\right)^3-4x^2\left(2x-1\right)\)

\(=\left(2x-1\right)\left[\left(2x-1\right)^2-4x^2\right]\)

\(=\left(2x-1\right)\left(2x-1-2x\right)\left(2x-1+2x\right)\)

\(=\left(1-2x\right)\left(4x-1\right)\)

c: \(x^2-6x+8=\left(x-2\right)\left(x-4\right)\)