Mister Exam
Factor y^4-12*y^2-8 squared
The solution
Factorization
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/ _______________\ / _______________\ / ______________\ / ______________\ | / ____ | | / ____ | | / ____ | | / ____ | \x + I*\/ -6 + 2*\/ 11 /*\x - I*\/ -6 + 2*\/ 11 /*\x + \/ 6 + 2*\/ 11 /*\x - \/ 6 + 2*\/ 11 /
$$\left(x - i \sqrt{-6 + 2 \sqrt{11}}\right) \left(x + i \sqrt{-6 + 2 \sqrt{11}}\right) \left(x + \sqrt{6 + 2 \sqrt{11}}\right) \left(x - \sqrt{6 + 2 \sqrt{11}}\right)$$
(((x + i*sqrt(-6 + 2*sqrt(11)))*(x - i*sqrt(-6 + 2*sqrt(11))))*(x + sqrt(6 + 2*sqrt(11))))*(x - sqrt(6 + 2*sqrt(11)))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 12 y^{2}\right) - 8$$
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
$$a = 1$$
$$b = -12$$
$$c = -8$$
Then
$$m = -6$$
$$n = -44$$
So,
$$\left(y^{2} - 6\right)^{2} - 44$$
$$\left(y^{4} - 12 y^{2}\right) - 8$$
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
$$a = 1$$
$$b = -12$$
$$c = -8$$
Then
$$m = -6$$
$$n = -44$$
So,
$$\left(y^{2} - 6\right)^{2} - 44$$
Combining rational expressions
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2 / 2\ -8 + y *\-12 + y /
$$y^{2} \left(y^{2} - 12\right) - 8$$
-8 + y^2*(-12 + y^2)