Mister Exam
Factor y^4-y^2+7 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - y^{2}\right) + 7$$
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
$$a = 1$$
$$b = -1$$
$$c = 7$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{27}{4}$$
$$\left(y^{4} - y^{2}\right) + 7$$
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
$$a = 1$$
$$b = -1$$
$$c = 7$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{27}{4}$$
Factorization
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/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | 4 ___ |atan\3*\/ 3 /| 4 ___ |atan\3*\/ 3 /|| | 4 ___ |atan\3*\/ 3 /| 4 ___ |atan\3*\/ 3 /|| | 4 ___ |atan\3*\/ 3 /| 4 ___ |atan\3*\/ 3 /|| | 4 ___ |atan\3*\/ 3 /| 4 ___ |atan\3*\/ 3 /|| |x + \/ 7 *cos|-------------| + I*\/ 7 *sin|-------------||*|x + \/ 7 *cos|-------------| - I*\/ 7 *sin|-------------||*|x + - \/ 7 *cos|-------------| + I*\/ 7 *sin|-------------||*|x + - \/ 7 *cos|-------------| - I*\/ 7 *sin|-------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right)$$
(((x + 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2))*(x + 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2))
Combining rational expressions
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2 / 2\ 7 + y *\-1 + y /
$$y^{2} \left(y^{2} - 1\right) + 7$$
7 + y^2*(-1 + y^2)