Mister Exam
Factor y^4+13*y^2-5 squared
The solution
Factorization
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/ _______________\ / _______________\ / _________________\ / _________________\ | / ____ | | / ____ | | / ____ | | / ____ | | / 13 3*\/ 21 | | / 13 3*\/ 21 | | / 13 3*\/ 21 | | / 13 3*\/ 21 | |x + I* / -- + -------- |*|x - I* / -- + -------- |*|x + / - -- + -------- |*|x - / - -- + -------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{13}{2} + \frac{3 \sqrt{21}}{2}}\right) \left(x + i \sqrt{\frac{13}{2} + \frac{3 \sqrt{21}}{2}}\right) \left(x + \sqrt{- \frac{13}{2} + \frac{3 \sqrt{21}}{2}}\right) \left(x - \sqrt{- \frac{13}{2} + \frac{3 \sqrt{21}}{2}}\right)$$
(((x + i*sqrt(13/2 + 3*sqrt(21)/2))*(x - i*sqrt(13/2 + 3*sqrt(21)/2)))*(x + sqrt(-13/2 + 3*sqrt(21)/2)))*(x - sqrt(-13/2 + 3*sqrt(21)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 13 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = 13$$
$$c = -5$$
Then
$$m = \frac{13}{2}$$
$$n = - \frac{189}{4}$$
So,
$$\left(y^{2} + \frac{13}{2}\right)^{2} - \frac{189}{4}$$
$$\left(y^{4} + 13 y^{2}\right) - 5$$
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
$$a = 1$$
$$b = 13$$
$$c = -5$$
Then
$$m = \frac{13}{2}$$
$$n = - \frac{189}{4}$$
So,
$$\left(y^{2} + \frac{13}{2}\right)^{2} - \frac{189}{4}$$
Combining rational expressions
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2 / 2\ -5 + y *\13 + y /
$$y^{2} \left(y^{2} + 13\right) - 5$$
-5 + y^2*(13 + y^2)