# How do you list all possible roots and find all factors of x^5+7x^3-3x-12?

Dec 23, 2016

Possible rational zeros are:

$\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 12$

but none are actually zeros.

This quintic is not solvable using radicals and elementary functions.

#### Explanation:

$f \left(x\right) = {x}^{5} + 7 {x}^{3} - 3 x - 12$

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Rational roots theorem

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 $- 12$ and $q$ a divisor of the coefficient $1$ of the leading term.

That means that the only possible rational zeros are:

$\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 12$

Note that when $\left\mid x \right\mid \ge 3$ we have:

$\left\mid 7 {x}^{3} \right\mid + \left\mid 3 x \right\mid + \left\mid 12 \right\mid \le \left\mid 7 {x}^{3} \right\mid + \left\mid 3 x \right\mid + \left\mid 4 x \right\mid = \left\mid 7 {x}^{3} \right\mid + \left\mid 7 x \right\mid < \left\mid 7 {x}^{3} \right\mid + \left\mid {x}^{3} \right\mid = \left\mid 8 {x}^{3} \right\mid < \left\mid {x}^{5} \right\mid$

So no $x$ with $\left\mid x \right\mid \ge 3$ can be a zero.

Checking the other possible rational zeros, we find:

$f \left(- 2\right) = - 32 - 56 + 6 - 12 = - 94$

$f \left(- 1\right) = - 1 - 7 + 3 - 12 = - 17$

$f \left(1\right) = 1 + 7 - 3 - 12 = - 7$

$f \left(2\right) = 32 + 7 - 6 - 12 = 21$

So $f \left(x\right)$ has no rational zeros, but has an irrational zero somewhere in $\left(1 , 2\right)$

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Quintic

In fact this is a typical quintic with $1$ Real zero and $4$ non-Real complex zeros, none of which are even expressible in terms of radicals and elementary functions - including trigonometric, exponential or logarithmic ones.

About the best you can do is find approximations using numerical methods such as Durand Kerner.

See https://socratic.org/s/aAGsRKkf for another example and a description of the Durand-Kerner algorithm for a quintic.

Using this algorithm, I found the following approximations:

${x}_{1} \approx 1.22622$

${x}_{2 , 3} \approx 0.101096 \pm 2.734 i$

${x}_{4 , 5} \approx - 0.714207 \pm 0.892944 i$

Here's the C++ program I used: 