# How do you find all zeros of the function 2x^6-3x^2-x+1?

Apr 2, 2016

See explanation...

#### Explanation:

By the rational roots theorem, the only possible rational zeros are expressible in the form $\frac{p}{q}$ with integers $p$ and $q$ with $p$ a divisor of the constant term $1$ and $q$ a divisor of the coefficient $2$ of the leading term.

That means that the only possible rational zeros are:

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

If we write $f \left(x\right) = {x}^{6} - 3 {x}^{2} - x + 1$ and evaluate $f \left(x\right)$ for these values, we find that none work. So $f \left(x\right)$ has no rational zeros.

Since $f \left(x\right)$ has degree $6$ it is not surprising to find that its roots have no simple closed algebraic formulation.

We can find them numerically, for example by using Newton's method.

$f ' \left(x\right) = 6 {x}^{5} - 6 x - 1$

If we choose an initial approximation ${a}_{0}$ for a zero of $f \left(x\right)$, then we can find better approximations by iterating using the formula:

${a}_{i + 1} = {a}_{i} - f \frac{x}{f ' \left(x\right)} = {a}_{i} - \frac{{x}^{6} - 3 {x}^{2} - x + 1}{6 {x}^{5} - 6 x - 1}$

The two Real zeros can readily be found by putting these formulae into a spreadsheet. To find the four Complex zeros is a little more complicated. If your spreadsheet application (like mine) does not handle Complex numbers directly, then you need to separate out Real and imaginary parts in separate columns.

If I have sufficient time, I may create such a spreadsheet.