Mister Exam
Factor y^4+3*y^2+2 squared
The solution
Factorization
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/ ___\ / ___\ (x + I)*(x - I)*\x + I*\/ 2 /*\x - I*\/ 2 /
$$\left(x - i\right) \left(x + i\right) \left(x + \sqrt{2} i\right) \left(x - \sqrt{2} i\right)$$
(((x + i)*(x - i))*(x + i*sqrt(2)))*(x - i*sqrt(2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 3 y^{2}\right) + 2$$
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
$$a = 1$$
$$b = 3$$
$$c = 2$$
Then
$$m = \frac{3}{2}$$
$$n = - \frac{1}{4}$$
So,
$$\left(y^{2} + \frac{3}{2}\right)^{2} - \frac{1}{4}$$
$$\left(y^{4} + 3 y^{2}\right) + 2$$
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
$$a = 1$$
$$b = 3$$
$$c = 2$$
Then
$$m = \frac{3}{2}$$
$$n = - \frac{1}{4}$$
So,
$$\left(y^{2} + \frac{3}{2}\right)^{2} - \frac{1}{4}$$
Combining rational expressions
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2 / 2\ 2 + y *\3 + y /
$$y^{2} \left(y^{2} + 3\right) + 2$$
2 + y^2*(3 + y^2)
Combinatorics
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/ 2\ / 2\ \1 + y /*\2 + y /
$$\left(y^{2} + 1\right) \left(y^{2} + 2\right)$$
(1 + y^2)*(2 + y^2)