Mister Exam

# Factor 5*p^4-3*p^2+13 squared

An expression to simplify:

### The solution

You have entered [src]
   4      2
5*p  - 3*p  + 13
$$\left(5 p^{4} - 3 p^{2}\right) + 13$$
5*p^4 - 3*p^2 + 13
Factorization [src]
/                   /    /  _____\\                    /    /  _____\\\ /                   /    /  _____\\                    /    /  _____\\\ /                     /    /  _____\\                    /    /  _____\\\ /                     /    /  _____\\                    /    /  _____\\\
|                   |    |\/ 251 ||                    |    |\/ 251 ||| |                   |    |\/ 251 ||                    |    |\/ 251 ||| |                     |    |\/ 251 ||                    |    |\/ 251 ||| |                     |    |\/ 251 ||                    |    |\/ 251 |||
|                   |atan|-------||                    |atan|-------||| |                   |atan|-------||                    |atan|-------||| |                     |atan|-------||                    |atan|-------||| |                     |atan|-------||                    |atan|-------|||
|     3/4 4 ____    |    \   3   /|      3/4 4 ____    |    \   3   /|| |     3/4 4 ____    |    \   3   /|      3/4 4 ____    |    \   3   /|| |       3/4 4 ____    |    \   3   /|      3/4 4 ____    |    \   3   /|| |       3/4 4 ____    |    \   3   /|      3/4 4 ____    |    \   3   /||
|    5   *\/ 13 *cos|-------------|   I*5   *\/ 13 *sin|-------------|| |    5   *\/ 13 *cos|-------------|   I*5   *\/ 13 *sin|-------------|| |      5   *\/ 13 *cos|-------------|   I*5   *\/ 13 *sin|-------------|| |      5   *\/ 13 *cos|-------------|   I*5   *\/ 13 *sin|-------------||
|                   \      2      /                    \      2      /| |                   \      2      /                    \      2      /| |                     \      2      /                    \      2      /| |                     \      2      /                    \      2      /|
|p + ------------------------------ + --------------------------------|*|p + ------------------------------ - --------------------------------|*|p + - ------------------------------ + --------------------------------|*|p + - ------------------------------ - --------------------------------|
\                  5                                 5                / \                  5                                 5                / \                    5                                 5                / \                    5                                 5                /
$$\left(p + \left(\frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(\frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(- \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(p + \left(- \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{13} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{251}}{3} \right)}}{2} \right)}}{5}\right)\right)$$
(((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,
$$5 \left(p^{2} - \frac{3}{10}\right)^{2} + \frac{251}{20}$$
General simplification [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
13.0 + 5.0*p^4 - 3.0*p^2
13.0 + 5.0*p^4 - 3.0*p^2
Powers [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
Trigonometric part [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
Rational denominator [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
Common denominator [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
Combining rational expressions [src]
      2 /        2\
13 + p *\-3 + 5*p /
$$p^{2} \left(5 p^{2} - 3\right) + 13$$
13 + p^2*(-3 + 5*p^2)
Combinatorics [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
Assemble expression [src]
        2      4
13 - 3*p  + 5*p 
$$5 p^{4} - 3 p^{2} + 13$$
13 - 3*p^2 + 5*p^4
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