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