Mister Exam
Factor -x^4+2*x^2-8 squared
The solution
Factorization
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$$\left(x + \left(2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right)$$
(((x + 2^(3/4)*cos(atan(sqrt(7))/2) + i*2^(3/4)*sin(atan(sqrt(7))/2))*(x + 2^(3/4)*cos(atan(sqrt(7))/2) - i*2^(3/4)*sin(atan(sqrt(7))/2)))*(x - 2^(3/4)*cos(atan(sqrt(7))/2) + i*2^(3/4)*sin(atan(sqrt(7))/2)))*(x - 2^(3/4)*cos(atan(sqrt(7))/2) - i*2^(3/4)*sin(atan(sqrt(7))/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- x^{4} + 2 x^{2}\right) - 8$$
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
$$a = -1$$
$$b = 2$$
$$c = -8$$
Then
$$m = -1$$
$$n = -7$$
So,
$$- \left(x^{2} - 1\right)^{2} - 7$$
$$\left(- x^{4} + 2 x^{2}\right) - 8$$
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
$$a = -1$$
$$b = 2$$
$$c = -8$$
Then
$$m = -1$$
$$n = -7$$
So,
$$- \left(x^{2} - 1\right)^{2} - 7$$
Combining rational expressions
[src]
2 / 2\ -8 + x *\2 - x /
$$x^{2} \left(2 - x^{2}\right) - 8$$
-8 + x^2*(2 - x^2)