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