Mister Exam
Factor y^4+4*y^2+2 squared
The solution
Factorization
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/ ___________\ / ___________\ / ___________\ / ___________\ | / ___ | | / ___ | | / ___ | | / ___ | \x + I*\/ 2 - \/ 2 /*\x - I*\/ 2 - \/ 2 /*\x + I*\/ 2 + \/ 2 /*\x - I*\/ 2 + \/ 2 /
$$\left(x - i \sqrt{2 - \sqrt{2}}\right) \left(x + i \sqrt{2 - \sqrt{2}}\right) \left(x + i \sqrt{\sqrt{2} + 2}\right) \left(x - i \sqrt{\sqrt{2} + 2}\right)$$
(((x + i*sqrt(2 - sqrt(2)))*(x - i*sqrt(2 - sqrt(2))))*(x + i*sqrt(2 + sqrt(2))))*(x - i*sqrt(2 + sqrt(2)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 4 y^{2}\right) + 2$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 4$$
$$c = 2$$
Then
$$m = 2$$
$$n = -2$$
So,
$$\left(y^{2} + 2\right)^{2} - 2$$
$$\left(y^{4} + 4 y^{2}\right) + 2$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 4$$
$$c = 2$$
Then
$$m = 2$$
$$n = -2$$
So,
$$\left(y^{2} + 2\right)^{2} - 2$$
Combining rational expressions
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2 / 2\ 2 + y *\4 + y /
$$y^{2} \left(y^{2} + 4\right) + 2$$
2 + y^2*(4 + y^2)