Mister Exam
Factor -y^4+8*y^2+9 squared
The solution
Factorization
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(x + 3)*(x - 3)*(x + I)*(x - I)
$$\left(x - 3\right) \left(x + 3\right) \left(x + i\right) \left(x - i\right)$$
(((x + 3)*(x - 3))*(x + i))*(x - i)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 8 y^{2}\right) + 9$$
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
$$a = -1$$
$$b = 8$$
$$c = 9$$
Then
$$m = -4$$
$$n = 25$$
So,
$$25 - \left(y^{2} - 4\right)^{2}$$
$$\left(- y^{4} + 8 y^{2}\right) + 9$$
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
$$a = -1$$
$$b = 8$$
$$c = 9$$
Then
$$m = -4$$
$$n = 25$$
So,
$$25 - \left(y^{2} - 4\right)^{2}$$
Combining rational expressions
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2 / 2\ 9 + y *\8 - y /
$$y^{2} \left(8 - y^{2}\right) + 9$$
9 + y^2*(8 - y^2)
Combinatorics
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/ 2\ -\1 + y /*(-3 + y)*(3 + y)
$$- \left(y - 3\right) \left(y + 3\right) \left(y^{2} + 1\right)$$
-(1 + y^2)*(-3 + y)*(3 + y)