### 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

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦