Mister Exam
Factor y^4-y^2+1 squared
The solution
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,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{3}{4}$$
Factorization
[src]
/ ___\ / ___\ / ___ \ / ___\ | I \/ 3 | | I \/ 3 | | \/ 3 I| | I \/ 3 | |x + - + -----|*|x + - - + -----|*|x + - ----- + -|*|x + - - - -----| \ 2 2 / \ 2 2 / \ 2 2/ \ 2 2 /
$$\left(x + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)\right) \left(x + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)\right) \left(x + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)\right) \left(x + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)\right)$$
(((x + i/2 + sqrt(3)/2)*(x - i/2 + sqrt(3)/2))*(x - sqrt(3)/2 + i/2))*(x - i/2 - sqrt(3)/2)
Combining rational expressions
[src]
2 / 2\ 1 + y *\-1 + y /
$$y^{2} \left(y^{2} - 1\right) + 1$$
1 + y^2*(-1 + y^2)