# How do you find all the real and complex roots of 2x^5-4x^4-4x^2+5=0?

Jul 6, 2016

Use a numerical method to find approximations:

$x \approx 2.2904$

$x \approx 0.925274$

$x \approx - 0.80773$

$x \approx - 0.203974 \pm 1.19116 i$

#### Explanation:

$f \left(x\right) = 2 {x}^{5} - 4 {x}^{4} - 4 {x}^{2} + 5$

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Fundamental theorem of algebra

The FTOA tells us that any non-zero polynomial of degree $> 0$ has a Complex (possibly Real) zero. A corollary of this, often quoted as part of the FTOA, is that a polynomial of degree $n > 0$ has exactly $n$ Complex (possibly Real) zeros counting multiplicity.

In our example $f \left(x\right)$ is of degree $5$, so has exactly $5$ zeros counting multiplicity.

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Rational root theorem

SInce $f \left(x\right)$ has integer coefficients, any rational zeros must be expressible in the for $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $5$ and $q$ a divisor of the coefficient $2$ of the leading term.

That means that the only possible rational zeros are:

$\pm \frac{1}{2} , \pm 1 , \pm \frac{5}{2} , \pm 5$

None of these satisfy $f \left(x\right) = 0$, so $f \left(x\right)$ has no rational zeros.

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Descartes rule of signs

The signs of the coefficients of $f \left(x\right)$ follow the pattern $+ - - +$. With two changes of signs, that means that $f \left(x\right)$ has $0$ or $2$ positive Real zeros.

The signs of the coefficients of $f \left(- x\right)$ follow the pattern $- - - +$. With one change of sign, that means that $f \left(x\right)$ has exactly $1$ negative Real zero.

That leaves $2$ or $4$ non-Real, Complex zeros, which will occur in Complex conjugate pairs.

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Quintic

In common with most quintics (and higher degree polynomials), the zeros of this one cannot be expressed using elementary functions like $n$th roots, or even trigonometric functions. About the best we can do is find approximations using a numerical method such as Durand-Kerner to find zeros:

$x \approx 2.2904$

$x \approx 0.925274$

$x \approx - 0.80773$

$x \approx - 0.203974 \pm 1.19116 i$

See https://socratic.org/s/avVFw8eC for more details of the method and another example quintic.