Mister Exam
Factor -y^4+8*y^2-4 squared
The solution
Factorization
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/ ___\ / ___\ / ___\ / ___\ \x + 1 - \/ 3 /*\x + -1 + \/ 3 /*\x + -1 - \/ 3 /*\x + 1 + \/ 3 /
$$\left(x + \left(-1 + \sqrt{3}\right)\right) \left(x + \left(1 - \sqrt{3}\right)\right) \left(x + \left(- \sqrt{3} - 1\right)\right) \left(x + \left(1 + \sqrt{3}\right)\right)$$
(((x + 1 - sqrt(3))*(x - 1 + sqrt(3)))*(x - 1 - sqrt(3)))*(x + 1 + sqrt(3))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 8 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = 8$$
$$c = -4$$
Then
$$m = -4$$
$$n = 12$$
So,
$$12 - \left(y^{2} - 4\right)^{2}$$
$$\left(- y^{4} + 8 y^{2}\right) - 4$$
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
$$a = -1$$
$$b = 8$$
$$c = -4$$
Then
$$m = -4$$
$$n = 12$$
So,
$$12 - \left(y^{2} - 4\right)^{2}$$
Combinatorics
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/ 2 \ / 2 \ -\-2 + y - 2*y/*\-2 + y + 2*y/
$$- \left(y^{2} - 2 y - 2\right) \left(y^{2} + 2 y - 2\right)$$
-(-2 + y^2 - 2*y)*(-2 + y^2 + 2*y)
Combining rational expressions
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2 / 2\ -4 + y *\8 - y /
$$y^{2} \left(8 - y^{2}\right) - 4$$
-4 + y^2*(8 - y^2)