Mister Exam
Factor y^4+13*y^2+1 squared
The solution
Factorization
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/ ______________\ / ______________\ / ______________\ / ______________\ | / _____ | | / _____ | | / _____ | | / _____ | | / 13 \/ 165 | | / 13 \/ 165 | | / 13 \/ 165 | | / 13 \/ 165 | |x + I* / -- - ------- |*|x - I* / -- - ------- |*|x + I* / -- + ------- |*|x - I* / -- + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{13}{2} - \frac{\sqrt{165}}{2}}\right) \left(x + i \sqrt{\frac{13}{2} - \frac{\sqrt{165}}{2}}\right) \left(x + i \sqrt{\frac{\sqrt{165}}{2} + \frac{13}{2}}\right) \left(x - i \sqrt{\frac{\sqrt{165}}{2} + \frac{13}{2}}\right)$$
(((x + i*sqrt(13/2 - sqrt(165)/2))*(x - i*sqrt(13/2 - sqrt(165)/2)))*(x + i*sqrt(13/2 + sqrt(165)/2)))*(x - i*sqrt(13/2 + sqrt(165)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 13 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = 13$$
$$c = 1$$
Then
$$m = \frac{13}{2}$$
$$n = - \frac{165}{4}$$
So,
$$\left(y^{2} + \frac{13}{2}\right)^{2} - \frac{165}{4}$$
$$\left(y^{4} + 13 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = 13$$
$$c = 1$$
Then
$$m = \frac{13}{2}$$
$$n = - \frac{165}{4}$$
So,
$$\left(y^{2} + \frac{13}{2}\right)^{2} - \frac{165}{4}$$
Combining rational expressions
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2 / 2\ 1 + y *\13 + y /
$$y^{2} \left(y^{2} + 13\right) + 1$$
1 + y^2*(13 + y^2)