# Exact roots of a polynomial: roots

### 5.27.22 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

  ⎤
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎦