Mister Exam
Factor -5*y^4-6*y^2-2 squared
The solution
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4 2 - 5*y - 6*y - 2
$$\left(- 5 y^{4} - 6 y^{2}\right) - 2$$
-5*y^4 - 6*y^2 - 2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- 5 y^{4} - 6 y^{2}\right) - 2$$
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
$$a = -5$$
$$b = -6$$
$$c = -2$$
Then
$$m = \frac{3}{5}$$
$$n = - \frac{1}{5}$$
So,
$$- 5 \left(y^{2} + \frac{3}{5}\right)^{2} - \frac{1}{5}$$
$$\left(- 5 y^{4} - 6 y^{2}\right) - 2$$
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
$$a = -5$$
$$b = -6$$
$$c = -2$$
Then
$$m = \frac{3}{5}$$
$$n = - \frac{1}{5}$$
So,
$$- 5 \left(y^{2} + \frac{3}{5}\right)^{2} - \frac{1}{5}$$
Factorization
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/ 4 ___ 3/4 /atan(1/3)\ 4 ___ 3/4 /atan(1/3)\\ / 4 ___ 3/4 /atan(1/3)\ 4 ___ 3/4 /atan(1/3)\\ / 4 ___ 3/4 /atan(1/3)\ 4 ___ 3/4 /atan(1/3)\\ / 4 ___ 3/4 /atan(1/3)\ 4 ___ 3/4 /atan(1/3)\\ | \/ 2 *5 *sin|---------| I*\/ 2 *5 *cos|---------|| | \/ 2 *5 *sin|---------| I*\/ 2 *5 *cos|---------|| | \/ 2 *5 *sin|---------| I*\/ 2 *5 *cos|---------|| | \/ 2 *5 *sin|---------| I*\/ 2 *5 *cos|---------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |x + ------------------------- + ---------------------------|*|x + ------------------------- - ---------------------------|*|x + - ------------------------- + ---------------------------|*|x + - ------------------------- - ---------------------------| \ 5 5 / \ 5 5 / \ 5 5 / \ 5 5 /
$$\left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{2} \right)}}{5}\right)\right)$$
(((x + 2^(1/4)*5^(3/4)*sin(atan(1/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(1/3)/2)/5)*(x + 2^(1/4)*5^(3/4)*sin(atan(1/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(1/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(1/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(1/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(1/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(1/3)/2)/5)
Combining rational expressions
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2 / 2\ -2 + y *\-6 - 5*y /
$$y^{2} \left(- 5 y^{2} - 6\right) - 2$$
-2 + y^2*(-6 - 5*y^2)