Mister Exam
Factor x^4-x^2-2 squared
The solution
Factorization
[src]
/ ___\ / ___\ \x + \/ 2 /*\x - \/ 2 /*(x + I)*(x - I)
$$\left(x - \sqrt{2}\right) \left(x + \sqrt{2}\right) \left(x + i\right) \left(x - i\right)$$
(((x + sqrt(2))*(x - sqrt(2)))*(x + i))*(x - i)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{4} - x^{2}\right) - 2$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
Then
$$m = - \frac{1}{2}$$
$$n = - \frac{9}{4}$$
So,
$$\left(x^{2} - \frac{1}{2}\right)^{2} - \frac{9}{4}$$
$$\left(x^{4} - x^{2}\right) - 2$$
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
$$a = 1$$
$$b = -1$$
$$c = -2$$
Then
$$m = - \frac{1}{2}$$
$$n = - \frac{9}{4}$$
So,
$$\left(x^{2} - \frac{1}{2}\right)^{2} - \frac{9}{4}$$
Combining rational expressions
[src]
2 / 2\ -2 + x *\-1 + x /
$$x^{2} \left(x^{2} - 1\right) - 2$$
-2 + x^2*(-1 + x^2)
Combinatorics
[src]
/ 2\ / 2\ \1 + x /*\-2 + x /
$$\left(x^{2} - 2\right) \left(x^{2} + 1\right)$$
(1 + x^2)*(-2 + x^2)