# Exact bounds for positive real roots of a polynomial: VAS_positive

### 5.29.4 Exact bounds for positive real roots of a polynomial: VAS_positive

The VAS_positive command uses the Vincent-Akritas-Strzebonski algorithm to find intervals containing the positive real roots of polynomials.

- VAS_positive takes one argument:
  P, a polynomial.
- VAS_positive(P) returns a list of intervals which contain the positive real roots of P, where each interval contains exactly one root.

#### Examples.

- Input:
  VAS_positive(x^3 - 7*x + 7)
  Output:

  ⎡
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎣

  1

  3
  2

  3
  2

  2

  ⎤
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎦

- Input:
  VAS_positive(x^5 + 2*x^4 - 6*x^3 - 7*x^2 + 7*x + 7)
  Output:

  ⎡
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎢
  ⎣

  1

  3
  2

  3
  2

  2

  ⎤
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎥
  ⎦

- Input:
  VAS_positive(x^3 - x^2 -2*x + 2)
  Output:

  ⎡ ⎣ | 1, | ⎡ ⎣ | 1,3 | ⎤ ⎦ | ⎤ ⎦