# How do you find all zeros of f(x)=5x^4+15x^2+10?

Dec 30, 2016

$f \left(x\right)$ has zeros $\pm i$ and $\pm \sqrt{2} i$

#### Explanation:

We will use the difference of squares identity, which can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = x$ and $b = i$ or $b = \sqrt{2} i$ as follows:

$f \left(x\right) = 5 {x}^{4} + 15 {x}^{2} + 10$

$\textcolor{w h i t e}{f \left(x\right)} = 5 \left({x}^{4} + 3 {x}^{2} + 2\right)$

$\textcolor{w h i t e}{f \left(x\right)} = 5 \left({x}^{2} + 1\right) \left({x}^{2} + 2\right)$

$\textcolor{w h i t e}{f \left(x\right)} = 5 \left({x}^{2} - {i}^{2}\right) \left({x}^{2} - {\left(\sqrt{2} i\right)}^{2}\right)$

$\textcolor{w h i t e}{f \left(x\right)} = 5 \left(x - i\right) \left(x + i\right) \left(x - \sqrt{2} i\right) \left(x + \sqrt{2} i\right)$

Hence the zeros of $f \left(x\right)$ are:

$x = \pm i$

$x = \pm \sqrt{2} i$