Mister Exam
Factor y^4+7*y^2-1 squared
The solution
Factorization
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/ ____________\ / ____________\ / ______________\ / ______________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 7 \/ 53 | | / 7 \/ 53 | | / 7 \/ 53 | | / 7 \/ 53 | |x + I* / - + ------ |*|x - I* / - + ------ |*|x + / - - + ------ |*|x - / - - + ------ | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{7}{2} + \frac{\sqrt{53}}{2}}\right) \left(x + i \sqrt{\frac{7}{2} + \frac{\sqrt{53}}{2}}\right) \left(x + \sqrt{- \frac{7}{2} + \frac{\sqrt{53}}{2}}\right) \left(x - \sqrt{- \frac{7}{2} + \frac{\sqrt{53}}{2}}\right)$$
(((x + i*sqrt(7/2 + sqrt(53)/2))*(x - i*sqrt(7/2 + sqrt(53)/2)))*(x + sqrt(-7/2 + sqrt(53)/2)))*(x - sqrt(-7/2 + sqrt(53)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 7 y^{2}\right) - 1$$
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
$$a = 1$$
$$b = 7$$
$$c = -1$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{53}{4}$$
So,
$$\left(y^{2} + \frac{7}{2}\right)^{2} - \frac{53}{4}$$
$$\left(y^{4} + 7 y^{2}\right) - 1$$
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
$$a = 1$$
$$b = 7$$
$$c = -1$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{53}{4}$$
So,
$$\left(y^{2} + \frac{7}{2}\right)^{2} - \frac{53}{4}$$
Combining rational expressions
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2 / 2\ -1 + y *\7 + y /
$$y^{2} \left(y^{2} + 7\right) - 1$$
-1 + y^2*(7 + y^2)