# How do you use the rational roots theorem to find all possible zeros of f(x) = 3X^5 - 2X^2 + 16X - 2?

Jun 22, 2016

Use a numerical method to find:

$x \approx 0.12701$

$x \approx 1.03966 \pm 0.998855 i$

$x \approx - 1.10317 \pm 1.14378 i$

#### Explanation:

$f \left(x\right) = 3 {x}^{5} - 2 {x}^{2} + 16 x - 2$

By the rational roots theorem, any rational zeros of $f \left(x\right)$ are expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $- 2$ and $q$ a divisor of the coefficient $3$ of the leading term.

That means that the only possible rational zeros of $f \left(x\right)$ are:

$\pm \frac{1}{3}$, $\pm \frac{2}{3}$, $\pm 1$, $\pm 2$

You can substitute each of these into the formula for $f \left(x\right)$ to find that none of them is a zero.

So this quintic has no rational zeros.

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By the fundamental theorem of algebra it does have exactly $5$ Complex, possibly Real, zeros, counting multiplicity.

In addition, since its coefficients are Real, any non-Real Complex zeros must occur in Complex conjugate pairs.

Typically for a quintic, the zeros of this particular one are not expressible in terms of $n$th roots.

About the best we can do is find numeric approximations using a method like Durand-Kerner. See https://socratic.org/s/avyDEG5X for a fuller explanation, but the Durand-Kerner algorithm finds approximations for all $5$ zeros at once.

The results I found were:

$x \approx 0.12701$

$x \approx 1.03966 \pm 0.998855 i$

$x \approx - 1.10317 \pm 1.14378 i$

Here's the C++ program I used to find them: