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