Mister Exam
Factor -y^4+8*y^2+1 squared
The solution
Factorization
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/ _____________\ / _____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + I*\/ -4 + \/ 17 /*\x - I*\/ -4 + \/ 17 /*\x + \/ 4 + \/ 17 /*\x - \/ 4 + \/ 17 /
$$\left(x - i \sqrt{-4 + \sqrt{17}}\right) \left(x + i \sqrt{-4 + \sqrt{17}}\right) \left(x + \sqrt{4 + \sqrt{17}}\right) \left(x - \sqrt{4 + \sqrt{17}}\right)$$
(((x + i*sqrt(-4 + sqrt(17)))*(x - i*sqrt(-4 + sqrt(17))))*(x + sqrt(4 + sqrt(17))))*(x - sqrt(4 + sqrt(17)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 8 y^{2}\right) + 1$$
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
$$a = -1$$
$$b = 8$$
$$c = 1$$
Then
$$m = -4$$
$$n = 17$$
So,
$$17 - \left(y^{2} - 4\right)^{2}$$
$$\left(- y^{4} + 8 y^{2}\right) + 1$$
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
$$a = -1$$
$$b = 8$$
$$c = 1$$
Then
$$m = -4$$
$$n = 17$$
So,
$$17 - \left(y^{2} - 4\right)^{2}$$
Combining rational expressions
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2 / 2\ 1 + y *\8 - y /
$$y^{2} \left(8 - y^{2}\right) + 1$$
1 + y^2*(8 - y^2)