Mister Exam
Factor -y^4-6*y^2+6 squared
The solution
Factorization
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/ ____________\ / ____________\ / _____________\ / _____________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + I*\/ 3 + \/ 15 /*\x - I*\/ 3 + \/ 15 /*\x + \/ -3 + \/ 15 /*\x - \/ -3 + \/ 15 /
$$\left(x - i \sqrt{3 + \sqrt{15}}\right) \left(x + i \sqrt{3 + \sqrt{15}}\right) \left(x + \sqrt{-3 + \sqrt{15}}\right) \left(x - \sqrt{-3 + \sqrt{15}}\right)$$
(((x + i*sqrt(3 + sqrt(15)))*(x - i*sqrt(3 + sqrt(15))))*(x + sqrt(-3 + sqrt(15))))*(x - sqrt(-3 + sqrt(15)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} - 6 y^{2}\right) + 6$$
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
$$a = -1$$
$$b = -6$$
$$c = 6$$
Then
$$m = 3$$
$$n = 15$$
So,
$$15 - \left(y^{2} + 3\right)^{2}$$
$$\left(- y^{4} - 6 y^{2}\right) + 6$$
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
$$a = -1$$
$$b = -6$$
$$c = 6$$
Then
$$m = 3$$
$$n = 15$$
So,
$$15 - \left(y^{2} + 3\right)^{2}$$
Combining rational expressions
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2 / 2\ 6 + y *\-6 - y /
$$y^{2} \left(- y^{2} - 6\right) + 6$$
6 + y^2*(-6 - y^2)