Question

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

Answer 1

Gathering common terms

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)\\

Further simplifying,

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)\\

Thus the final factorized expression is

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