Question

factorise 1+X+Y-Z+XY-YZ-ZX-XYZ

Answer 1

The given expression is $1+X+Y-Z+XY-YZ-ZX-XYZ$.

Gathering common terms

[tex]1+X+Y-Z+XY-YZ-ZX-XYZ=1+X+Y-YZ-ZX-Z+XY-XYZ\\ 1+X+Y-Z+XY-YZ-ZX-XYZ=1-Z+X+Y-Z(Y+X)+XY-XYZ\\ 1+X+Y-Z+XY-YZ-ZX-XYZ=1-Z+(X+Y)-Z(Y+X)+(XY-XYZ)\\ 1+X+Y-Z+XY-YZ-ZX-XYZ=(1-Z)+(X+Y)(1-Z)+XY(1-Z)\\[/tex]

Further simplifying,

[tex]1+X+Y-Z+XY-YZ-ZX-XYZ=(1-Z)[1+(X+Y)+XY]\\ 1+X+Y-Z+XY-YZ-ZX-XYZ=(1-Z)(1+X)(1+Y)\\[/tex]

Thus the final factorized expression is

$1+X+Y-Z+XY-YZ-ZX-XYZ=(1+X)(1+Y)(1-Z)$