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