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