### 2.5.5 Complex numbers

When factoring polynomials (see Section 9.1.10), by default Xcas won’t introduce complex numbers if they aren’t already being used. For example,

factor(x^2+2)

simply returns

x2+2 |  |  |  |  |  |  |  |  |  |

but if an expression already involves complex numbers then Xcas uses them:

factor(i*x^2+2*i)

| ⎛ ⎜ ⎝ | x−i | √ |
2
| ⎞ ⎟ ⎠ | ⎛ ⎜ ⎝ | i x− | √ |
2
| ⎞ ⎟ ⎠

Xcas can also find complex roots when complex numbers are not present; for example, the command cfactor (see Section 9.1.10) or cFactor command factors over the complex numbers. For example:

cfactor(x^2+2)

| ⎛ ⎜ ⎝ | x+i | √ |
2
| ⎞ ⎟ ⎠ | ⎛ ⎜ ⎝ | x−i | √ |
2
| ⎞ ⎟ ⎠

If you want Xcas to use complex numbers by default, you can turn on complex mode. In complex mode, the command line

factor(x^2+2)

returns

| ⎛ ⎜ ⎝ | x+i | √ |
2
| ⎞ ⎟ ⎠ | ⎛ ⎜ ⎝ | x−i | √ |
2
| ⎞ ⎟ ⎠

You can turn on complex mode from the CAS configuration screen (see Section 2.5.7). This mode is determined by the value of the variable complex_mode; if this is 1 then complex mode is on, if this is 0 then complex mode is off. This option will be stored in the configuration file (see Section 2.5.10), and so can also be set there.