### 11.1.22 Exact roots of a polynomial

The roots command finds roots of polynomials with their multiplicities.

- roots takes one mandatory and one optional argument:
  - P, a symbolic polynomial expression.
  - Optionally, x, the name of the variable (the default is x).
- roots(P ⟨,x⟩) returns a 2 column matrix: each row is the list consisting of a root of P and its multiplicity.

#### Examples

Find the roots of P(x)=x5−2x4+x3.

roots(x^5-16*x^4+x^3)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
3 | √ |
7
| +8
| 1
−3 | √ |
7
| +8
| 1
0 | 3
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Find the roots of x10−15x8+90x6−270x4+405x2−243=(x2−3)5.

roots(x^10-15*x^8+90*x^6-270*x^4+405*x^2-243)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
√ |
3
| 5
− | √ |
3
| 5
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Find the roots of t3−1.

roots(t^3-1,t)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
1 | 1
i | √ |
3
| −1
2
| 1
−i | √ |
3
| −1
2
| 1
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦