Mister Exam
Factor -y^4+5*y^2-8 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + 5 y^{2}\right) - 8$$
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
$$a = -1$$
$$b = 5$$
$$c = -8$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{7}{4}$$
So,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{7}{4}$$
$$\left(- y^{4} + 5 y^{2}\right) - 8$$
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
$$a = -1$$
$$b = 5$$
$$c = -8$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{7}{4}$$
So,
$$- \left(y^{2} - \frac{5}{2}\right)^{2} - \frac{7}{4}$$
Factorization
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$$\left(x + \left(2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{5} \right)}}{2} \right)}\right)\right)$$
(((x + 2^(3/4)*cos(atan(sqrt(7)/5)/2) + i*2^(3/4)*sin(atan(sqrt(7)/5)/2))*(x + 2^(3/4)*cos(atan(sqrt(7)/5)/2) - i*2^(3/4)*sin(atan(sqrt(7)/5)/2)))*(x - 2^(3/4)*cos(atan(sqrt(7)/5)/2) + i*2^(3/4)*sin(atan(sqrt(7)/5)/2)))*(x - 2^(3/4)*cos(atan(sqrt(7)/5)/2) - i*2^(3/4)*sin(atan(sqrt(7)/5)/2))
Combining rational expressions
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2 / 2\ -8 + y *\5 - y /
$$y^{2} \left(5 - y^{2}\right) - 8$$
-8 + y^2*(5 - y^2)