Question

Use the rational root theorem to list all possible rational roots for the equation.

3x^3+9x-6=0

Answer 1

All possible rational roots are: 1, 1/3, 2, 2/3, 3, 6.

Explanation:

The given expression is

`3x^(3)+9x-6=0`

The rational root theorem states the divisors of the constant are possible roots. Additionally, if the term with the greater exponent as a coefficient different than one, then we must incluce its divisor, to then divide them by the divisor of the constant.

According to the theorem, all possible rational roots are

Divisors of 6: 1, 2, 3, 6.

Divisors of 3: 1, 3.

Possible rational roots: 1/1, 1/3, 2/1, 2/3, 3/1, 3/3, 6/1, 6/3.

Therefore, all possible rational roots are: 1, 1/3, 2, 2/3, 3, 6.

Answer 2

The possible rational roots are

`\pm 1, \pm 2,\pm3,\pm6,\pm (1)/(3),\pm (2)/(3)`

Explanation:

We have been given the equation 3x^3+9x-6=0 and we have to list all possible rational roots by rational root theorem.

The factors of constant term are
`1, 2, 3,6`

The factors of leading coefficient are
`1,3,`

From ration root theorem, the possible roots are the ratio of the factors of the constant term and the factors of the leading coefficient. We include both positive as well as negative, hence we must include plus minus.

`\pm(1,2,3,6)/(1,3)\\\\ =\pm 1, \pm 2,\pm3,\pm6,\pm (1)/(3),\pm (2)/(3)`