Mister Exam
Factor -y^4+y^2+11 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} + y^{2}\right) + 11$$
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
$$a = -1$$
$$b = 1$$
$$c = 11$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{45}{4}$$
So,
$$\frac{45}{4} - \left(y^{2} - \frac{1}{2}\right)^{2}$$
$$\left(- y^{4} + y^{2}\right) + 11$$
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
$$a = -1$$
$$b = 1$$
$$c = 11$$
Then
$$m = - \frac{1}{2}$$
$$n = \frac{45}{4}$$
So,
$$\frac{45}{4} - \left(y^{2} - \frac{1}{2}\right)^{2}$$
Factorization
[src]
/ _______________\ / _______________\ / _____________\ / _____________\ | / ___ | | / ___ | | / ___ | | / ___ | | / 1 3*\/ 5 | | / 1 3*\/ 5 | | / 1 3*\/ 5 | | / 1 3*\/ 5 | |x + I* / - - + ------- |*|x - I* / - - + ------- |*|x + / - + ------- |*|x - / - + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - i \sqrt{- \frac{1}{2} + \frac{3 \sqrt{5}}{2}}\right) \left(x + i \sqrt{- \frac{1}{2} + \frac{3 \sqrt{5}}{2}}\right) \left(x + \sqrt{\frac{1}{2} + \frac{3 \sqrt{5}}{2}}\right) \left(x - \sqrt{\frac{1}{2} + \frac{3 \sqrt{5}}{2}}\right)$$
(((x + i*sqrt(-1/2 + 3*sqrt(5)/2))*(x - i*sqrt(-1/2 + 3*sqrt(5)/2)))*(x + sqrt(1/2 + 3*sqrt(5)/2)))*(x - sqrt(1/2 + 3*sqrt(5)/2))
Combining rational expressions
[src]
2 / 2\ 11 + y *\1 - y /
$$y^{2} \left(1 - y^{2}\right) + 11$$
11 + y^2*(1 - y^2)