Mister Exam
Factor y^4-15*y^2+1 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - 15 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = -15$$
$$c = 1$$
Then
$$m = - \frac{15}{2}$$
$$n = - \frac{221}{4}$$
So,
$$\left(y^{2} - \frac{15}{2}\right)^{2} - \frac{221}{4}$$
$$\left(y^{4} - 15 y^{2}\right) + 1$$
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
$$a = 1$$
$$b = -15$$
$$c = 1$$
Then
$$m = - \frac{15}{2}$$
$$n = - \frac{221}{4}$$
So,
$$\left(y^{2} - \frac{15}{2}\right)^{2} - \frac{221}{4}$$
Factorization
[src]
/ ______________\ / ______________\ / ______________\ / ______________\ | / _____ | | / _____ | | / _____ | | / _____ | | / 15 \/ 221 | | / 15 \/ 221 | | / 15 \/ 221 | | / 15 \/ 221 | |x + / -- - ------- |*|x - / -- - ------- |*|x + / -- + ------- |*|x - / -- + ------- | \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
$$\left(x - \sqrt{\frac{15}{2} - \frac{\sqrt{221}}{2}}\right) \left(x + \sqrt{\frac{15}{2} - \frac{\sqrt{221}}{2}}\right) \left(x + \sqrt{\frac{\sqrt{221}}{2} + \frac{15}{2}}\right) \left(x - \sqrt{\frac{\sqrt{221}}{2} + \frac{15}{2}}\right)$$
(((x + sqrt(15/2 - sqrt(221)/2))*(x - sqrt(15/2 - sqrt(221)/2)))*(x + sqrt(15/2 + sqrt(221)/2)))*(x - sqrt(15/2 + sqrt(221)/2))
Combining rational expressions
[src]
2 / 2\ 1 + y *\-15 + y /
$$y^{2} \left(y^{2} - 15\right) + 1$$
1 + y^2*(-15 + y^2)