Question

Answer 1

a) \(x^3-x\)

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

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

b) \(\left(2x+y\right)^2-4y^2\)

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

\(=\left(2x-y\right)\left(2x+3y\right)\)

c) \(x^3+y^3+x+y\)

\(=\left(x^3+y^3\right)+\left(x+y\right)\)

\(=\left(x+y\right)\left(x^2+xy+y^2\right)+\left(x+y\right)\)

\(=\left(x+y\right)\left(x^2+xy+y^2+1\right)\)

d) \(x^3+4x^2y+4xy^2-9x\)

\(=x\left(x^2+4xy+4y^2-9\right)\)

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

\(=x\left(x+2y-3\right)\left(x+2y+3\right)\)

e) \(x^2+4-y^2+4x\)

\(=\left(x^2+4x+4\right)-y^2\)

\(=\left(x+2\right)^2-y^2\)

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

f) \(x^4+4\)

\(=x^4+2x^3+2x^2-2x^3-4x^2-4x+2x^2+4x+4\)

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

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

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