The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{4} + x^{2}\right) + 1$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 1$$
$$c = 1$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{3}{4}$$
So,
$$\left(x^{2} + \frac{1}{2}\right)^{2} + \frac{3}{4}$$
/ ___\ / ___\ / ___\ / ___\
| 1 I*\/ 3 | | 1 I*\/ 3 | | 1 I*\/ 3 | | 1 I*\/ 3 |
|x + - + -------|*|x + - - -------|*|x + - - + -------|*|x + - - - -------|
\ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right)$$
(((x + 1/2 + i*sqrt(3)/2)*(x + 1/2 - i*sqrt(3)/2))*(x - 1/2 + i*sqrt(3)/2))*(x - 1/2 - i*sqrt(3)/2)
General simplification
[src]
$$x^{4} + x^{2} + 1$$
Assemble expression
[src]
$$x^{4} + x^{2} + 1$$
Rational denominator
[src]
$$x^{4} + x^{2} + 1$$
/ 2\ / 2 \
\1 + x + x /*\1 + x - x/
$$\left(x^{2} - x + 1\right) \left(x^{2} + x + 1\right)$$
(1 + x + x^2)*(1 + x^2 - x)
Combining rational expressions
[src]
$$x^{2} \left(x^{2} + 1\right) + 1$$