Mister Exam
Factor y^4-9*y^2-10 squared
The solution
Factorization
[src]
/ ____\ / ____\ \x + \/ 10 /*\x - \/ 10 /*(x + I)*(x - I)
$$\left(x - \sqrt{10}\right) \left(x + \sqrt{10}\right) \left(x + i\right) \left(x - i\right)$$
(((x + sqrt(10))*(x - sqrt(10)))*(x + i))*(x - i)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 9 y^{2}\right) - 10$$
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
$$a = 1$$
$$b = -9$$
$$c = -10$$
Then
$$m = - \frac{9}{2}$$
$$n = - \frac{121}{4}$$
So,
$$\left(y^{2} - \frac{9}{2}\right)^{2} - \frac{121}{4}$$
$$\left(y^{4} - 9 y^{2}\right) - 10$$
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
$$a = 1$$
$$b = -9$$
$$c = -10$$
Then
$$m = - \frac{9}{2}$$
$$n = - \frac{121}{4}$$
So,
$$\left(y^{2} - \frac{9}{2}\right)^{2} - \frac{121}{4}$$
Combining rational expressions
[src]
2 / 2\ -10 + y *\-9 + y /
$$y^{2} \left(y^{2} - 9\right) - 10$$
-10 + y^2*(-9 + y^2)
Combinatorics
[src]
/ 2\ / 2\ \1 + y /*\-10 + y /
$$\left(y^{2} - 10\right) \left(y^{2} + 1\right)$$
(1 + y^2)*(-10 + y^2)