Mister Exam
Factor y^4+7*y^2+1 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 7 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = 7$$
$$c = 1$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{45}{4}$$
So,
$$\left(y^{2} + \frac{7}{2}\right)^{2} - \frac{45}{4}$$
$$\left(y^{4} + 7 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = 7$$
$$c = 1$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{45}{4}$$
So,
$$\left(y^{2} + \frac{7}{2}\right)^{2} - \frac{45}{4}$$
Factorization
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/ _____________\ / _____________\ / _____________\ / _____________\ | / ___ | | / ___ | | / ___ | | / ___ | | / 7 3*\/ 5 | | / 7 3*\/ 5 | | / 7 3*\/ 5 | | / 7 3*\/ 5 | |x + I* / - - ------- |*|x - I* / - - ------- |*|x + I* / - + ------- |*|x - I* / - + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{\frac{7}{2} - \frac{3 \sqrt{5}}{2}}\right) \left(x + i \sqrt{\frac{7}{2} - \frac{3 \sqrt{5}}{2}}\right) \left(x + i \sqrt{\frac{3 \sqrt{5}}{2} + \frac{7}{2}}\right) \left(x - i \sqrt{\frac{3 \sqrt{5}}{2} + \frac{7}{2}}\right)$$
(((x + i*sqrt(7/2 - 3*sqrt(5)/2))*(x - i*sqrt(7/2 - 3*sqrt(5)/2)))*(x + i*sqrt(7/2 + 3*sqrt(5)/2)))*(x - i*sqrt(7/2 + 3*sqrt(5)/2))
Combining rational expressions
[src]
2 / 2\ 1 + y *\7 + y /
$$y^{2} \left(y^{2} + 7\right) + 1$$
1 + y^2*(7 + y^2)