Mister Exam
Factor 8*y^4+6*y^2+7 squared
The solution
Factorization
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$$\left(x + \left(\frac{\sqrt[4]{14} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{14} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(\frac{\sqrt[4]{14} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{14} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{\sqrt[4]{14} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{14} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{\sqrt[4]{14} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{14} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{47}}{3} \right)}}{2} \right)}}{2}\right)\right)$$
(((x + 14^(1/4)*sin(atan(sqrt(47)/3)/2)/2 + i*14^(1/4)*cos(atan(sqrt(47)/3)/2)/2)*(x + 14^(1/4)*sin(atan(sqrt(47)/3)/2)/2 - i*14^(1/4)*cos(atan(sqrt(47)/3)/2)/2))*(x - 14^(1/4)*sin(atan(sqrt(47)/3)/2)/2 + i*14^(1/4)*cos(atan(sqrt(47)/3)/2)/2))*(x - 14^(1/4)*sin(atan(sqrt(47)/3)/2)/2 - i*14^(1/4)*cos(atan(sqrt(47)/3)/2)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(8 y^{4} + 6 y^{2}\right) + 7$$
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
$$a = 8$$
$$b = 6$$
$$c = 7$$
Then
$$m = \frac{3}{8}$$
$$n = \frac{47}{8}$$
So,
$$8 \left(y^{2} + \frac{3}{8}\right)^{2} + \frac{47}{8}$$
$$\left(8 y^{4} + 6 y^{2}\right) + 7$$
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
$$a = 8$$
$$b = 6$$
$$c = 7$$
Then
$$m = \frac{3}{8}$$
$$n = \frac{47}{8}$$
So,
$$8 \left(y^{2} + \frac{3}{8}\right)^{2} + \frac{47}{8}$$
Combining rational expressions
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2 / 2\ 7 + 2*y *\3 + 4*y /
$$2 y^{2} \left(4 y^{2} + 3\right) + 7$$
7 + 2*y^2*(3 + 4*y^2)