Question

What are the 8 possible rational roots?

Answer 1

p/q = {-6, -3, -2, -1, 1, 2, 3, 6}

Explanation:

You want the possible rational roots of x⁴ +2x³ -5x² -x +6.

Rational root theorem

The rational root theorem tells you any rational roots will be of the form ...

±(divisor of the constant)/(divisor of the leading coefficient)

Here, the leading coefficient is 1, so the possible rational roots are ...

p/q = ±divisor of 6

p/q = {-6, -3, -2, -1, 1, 2, 3, 6}

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Additional comment

In fact, this quartic has no rational roots. It has two negative irrational real roots, and two complex roots.

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