Mister Exam
Factor y^4-5*y^2+15 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 5 y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -5$$
$$c = 15$$
Then
$$m = - \frac{5}{2}$$
$$n = \frac{35}{4}$$
So,
$$\left(y^{2} - \frac{5}{2}\right)^{2} + \frac{35}{4}$$
$$\left(y^{4} - 5 y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -5$$
$$c = 15$$
Then
$$m = - \frac{5}{2}$$
$$n = \frac{35}{4}$$
So,
$$\left(y^{2} - \frac{5}{2}\right)^{2} + \frac{35}{4}$$
Factorization
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/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ | | |\/ 35 || | |\/ 35 ||| | | |\/ 35 || | |\/ 35 ||| | | |\/ 35 || | |\/ 35 ||| | | |\/ 35 || | |\/ 35 ||| | |atan|------|| |atan|------||| | |atan|------|| |atan|------||| | |atan|------|| |atan|------||| | |atan|------|| |atan|------||| | 4 ____ | \ 5 /| 4 ____ | \ 5 /|| | 4 ____ | \ 5 /| 4 ____ | \ 5 /|| | 4 ____ | \ 5 /| 4 ____ | \ 5 /|| | 4 ____ | \ 5 /| 4 ____ | \ 5 /|| |x + \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + \/ 15 *cos|------------| - I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| - I*\/ 15 *sin|------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right)$$
(((x + 15^(1/4)*cos(atan(sqrt(35)/5)/2) + i*15^(1/4)*sin(atan(sqrt(35)/5)/2))*(x + 15^(1/4)*cos(atan(sqrt(35)/5)/2) - i*15^(1/4)*sin(atan(sqrt(35)/5)/2)))*(x - 15^(1/4)*cos(atan(sqrt(35)/5)/2) + i*15^(1/4)*sin(atan(sqrt(35)/5)/2)))*(x - 15^(1/4)*cos(atan(sqrt(35)/5)/2) - i*15^(1/4)*sin(atan(sqrt(35)/5)/2))
Combining rational expressions
[src]
2 / 2\ 15 + y *\-5 + y /
$$y^{2} \left(y^{2} - 5\right) + 15$$
15 + y^2*(-5 + y^2)