Mister Exam
Factor x^4+x^2-2 squared
The solution
Factorization
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/ ___\ / ___\ (x + 1)*(x - 1)*\x + I*\/ 2 /*\x - I*\/ 2 /
$$\left(x - 1\right) \left(x + 1\right) \left(x + \sqrt{2} i\right) \left(x - \sqrt{2} i\right)$$
(((x + 1)*(x - 1))*(x + i*sqrt(2)))*(x - i*sqrt(2))
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,
$$\left(x^{2} + \frac{1}{2}\right)^{2} - \frac{9}{4}$$
Combinatorics
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/ 2\ (1 + x)*(-1 + x)*\2 + x /
$$\left(x - 1\right) \left(x + 1\right) \left(x^{2} + 2\right)$$
(1 + x)*(-1 + x)*(2 + x^2)
Combining rational expressions
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2 / 2\ -2 + x *\1 + x /
$$x^{2} \left(x^{2} + 1\right) - 2$$
-2 + x^2*(1 + x^2)