Mister Exam
Factor y^4-y^2-1 squared
The solution
Factorization
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/ _____________\ / _____________\ / ___________\ / ___________\ | / ___ | | / ___ | | / ___ | | / ___ | | / 1 \/ 5 | | / 1 \/ 5 | | / 1 \/ 5 | | / 1 \/ 5 | |x + I* / - - + ----- |*|x - I* / - - + ----- |*|x + / - + ----- |*|x - / - + ----- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{- \frac{1}{2} + \frac{\sqrt{5}}{2}}\right) \left(x + i \sqrt{- \frac{1}{2} + \frac{\sqrt{5}}{2}}\right) \left(x + \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}\right) \left(x - \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}\right)$$
(((x + i*sqrt(-1/2 + sqrt(5)/2))*(x - i*sqrt(-1/2 + sqrt(5)/2)))*(x + sqrt(1/2 + sqrt(5)/2)))*(x - sqrt(1/2 + sqrt(5)/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
$$m = - \frac{1}{2}$$
$$n = - \frac{5}{4}$$
So,
$$\left(y^{2} - \frac{1}{2}\right)^{2} - \frac{5}{4}$$
$$\left(y^{4} - y^{2}\right) - 1$$
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
$$m = - \frac{1}{2}$$
$$n = - \frac{5}{4}$$
So,
$$\left(y^{2} - \frac{1}{2}\right)^{2} - \frac{5}{4}$$
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)