Question

What are all the rational zeros of `2x^3-15x^2+9x+22`?

Answer 1

Use the rational roots theorem to find the possible rational zeros.

Explanation:

`f(x) = 2x^3-15x^2+9x+22`

By the rational roots theorem, the only possible rational zeros are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `22` and `q` a divisor of the coefficient `2` of the leading term.

So the only possible rational zeros are:

`+-1/2, +-1, +-2, +-11/2, +-11, +-22`

Evaluating `f(x)` for each of these we find that none work, so `f(x)` has no rational zeros.

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We can find out a little more without actually solving the cubic...

The discriminant `Delta` of a cubic polynomial in the form `ax^3+bx^2+cx+d` is given by the formula:

`Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd`

In our example, `a=2`, `b=-15`, `c=9` and `d=22`, so we find:

`Delta = 18225-5832+297000-52272-106920 = 150201`

Since `Delta > 0` this cubic has `3` Real zeros.

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Using Descartes' rule of signs, we can determine that two of these zeros are positive and one negative.