Mister Exam
Factor 5*p^4-3*p^2+13 squared
The solution
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4 2 5*p - 3*p + 13
$$\left(5 p^{4} - 3 p^{2}\right) + 13$$
5*p^4 - 3*p^2 + 13
Factorization
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$$\left(p + \left(\frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(\frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(- \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(- \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right)$$
(((p + 5^(3/4)*13^(1/4)*cos(atan(sqrt(251)/3)/2)/5 + i*5^(3/4)*13^(1/4)*sin(atan(sqrt(251)/3)/2)/5)*(p + 5^(3/4)*13^(1/4)*cos(atan(sqrt(251)/3)/2)/5 - i*5^(3/4)*13^(1/4)*sin(atan(sqrt(251)/3)/2)/5))*(p - 5^(3/4)*13^(1/4)*cos(atan(sqrt(251)/3)/2)/5 + i*5^(3/4)*13^(1/4)*sin(atan(sqrt(251)/3)/2)/5))*(p - 5^(3/4)*13^(1/4)*cos(atan(sqrt(251)/3)/2)/5 - i*5^(3/4)*13^(1/4)*sin(atan(sqrt(251)/3)/2)/5)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(5 p^{4} - 3 p^{2}\right) + 13$$
To do this, let's use the formula
$$a p^{4} + b p^{2} + c = a \left(m + p^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 5$$
$$b = -3$$
$$c = 13$$
Then
$$m = - \frac{3}{10}$$
$$n = \frac{251}{20}$$
So,
$$5 \left(p^{2} - \frac{3}{10}\right)^{2} + \frac{251}{20}$$
$$\left(5 p^{4} - 3 p^{2}\right) + 13$$
To do this, let's use the formula
$$a p^{4} + b p^{2} + c = a \left(m + p^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 5$$
$$b = -3$$
$$c = 13$$
Then
$$m = - \frac{3}{10}$$
$$n = \frac{251}{20}$$
So,
$$5 \left(p^{2} - \frac{3}{10}\right)^{2} + \frac{251}{20}$$
Combining rational expressions
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2 / 2\ 13 + p *\-3 + 5*p /
$$p^{2} \left(5 p^{2} - 3\right) + 13$$
13 + p^2*(-3 + 5*p^2)