Mister Exam

# Factor 2*p^4+p^2-2 squared

An expression to simplify:

### The solution

You have entered [src]
   4    2
2*p  + p  - 2
$$\left(2 p^{4} + p^{2}\right) - 2$$
2*p^4 + p^2 - 2
General simplification [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
Factorization [src]
/           ____________\ /           ____________\ /         ______________\ /         ______________\
|          /       ____ | |          /       ____ | |        /         ____ | |        /         ____ |
|         /  1   \/ 17  | |         /  1   \/ 17  | |       /    1   \/ 17  | |       /    1   \/ 17  |
|p + I*  /   - + ------ |*|p - I*  /   - + ------ |*|p +   /   - - + ------ |*|p -   /   - - + ------ |
\      \/    4     4    / \      \/    4     4    / \    \/      4     4    / \    \/      4     4    /
$$\left(p - i \sqrt{\frac{1}{4} + \frac{\sqrt{17}}{4}}\right) \left(p + i \sqrt{\frac{1}{4} + \frac{\sqrt{17}}{4}}\right) \left(p + \sqrt{- \frac{1}{4} + \frac{\sqrt{17}}{4}}\right) \left(p - \sqrt{- \frac{1}{4} + \frac{\sqrt{17}}{4}}\right)$$
(((p + i*sqrt(1/4 + sqrt(17)/4))*(p - i*sqrt(1/4 + sqrt(17)/4)))*(p + sqrt(-1/4 + sqrt(17)/4)))*(p - sqrt(-1/4 + sqrt(17)/4))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(2 p^{4} + p^{2}\right) - 2$$
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
$$a = 2$$
$$b = 1$$
$$c = -2$$
Then
$$m = \frac{1}{4}$$
$$n = - \frac{17}{8}$$
So,
$$2 \left(p^{2} + \frac{1}{4}\right)^{2} - \frac{17}{8}$$
Assemble expression [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
-2.0 + p^2 + 2.0*p^4
-2.0 + p^2 + 2.0*p^4
Powers [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
Combinatorics [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
Common denominator [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
Combining rational expressions [src]
      2 /       2\
-2 + p *\1 + 2*p /
$$p^{2} \left(2 p^{2} + 1\right) - 2$$
-2 + p^2*(1 + 2*p^2)
Rational denominator [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4
Trigonometric part [src]
      2      4
-2 + p  + 2*p 
$$2 p^{4} + p^{2} - 2$$
-2 + p^2 + 2*p^4