Exact roots of a polynomial: roots

### 6.27.22 Exact roots of a polynomial: roots

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.
  Input:
  roots(x^5-16*x^4+x^3)
  Output:

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

- Find the roots of x10−15x8+90x6−270x4+405x2−243=(x2−3)5.
  Input:
  roots(x^10-15*x^8+90*x^6-270*x^4+405*x^2-243)
  Output:

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

- Find the roots of t3−1.
  Input:
  roots(t^3-1,t)
  Output:

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