Mister Exam
Factor -t^4+10*t^2+1 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- t^{4} + 10 t^{2}\right) + 1$$
To do this, let's use the formula
$$a t^{4} + b t^{2} + c = a \left(m + t^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 10$$
$$c = 1$$
Then
$$m = -5$$
$$n = 26$$
So,
$$26 - \left(t^{2} - 5\right)^{2}$$
$$\left(- t^{4} + 10 t^{2}\right) + 1$$
To do this, let's use the formula
$$a t^{4} + b t^{2} + c = a \left(m + t^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = 10$$
$$c = 1$$
Then
$$m = -5$$
$$n = 26$$
So,
$$26 - \left(t^{2} - 5\right)^{2}$$
Factorization
[src]
/ _____________\ / _____________\ / ____________\ / ____________\ | / ____ | | / ____ | | / ____ | | / ____ | \t + I*\/ -5 + \/ 26 /*\t - I*\/ -5 + \/ 26 /*\t + \/ 5 + \/ 26 /*\t - \/ 5 + \/ 26 /
$$\left(t - i \sqrt{-5 + \sqrt{26}}\right) \left(t + i \sqrt{-5 + \sqrt{26}}\right) \left(t + \sqrt{5 + \sqrt{26}}\right) \left(t - \sqrt{5 + \sqrt{26}}\right)$$
(((t + i*sqrt(-5 + sqrt(26)))*(t - i*sqrt(-5 + sqrt(26))))*(t + sqrt(5 + sqrt(26))))*(t - sqrt(5 + sqrt(26)))
Combining rational expressions
[src]
2 / 2\ 1 + t *\10 - t /
$$t^{2} \left(10 - t^{2}\right) + 1$$
1 + t^2*(10 - t^2)