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