Mister Exam

# Factor -y^4-y^2-3 squared

An expression to simplify:

### The solution

You have entered [src]
   4    2
- y  - y  - 3
$$\left(- y^{4} - y^{2}\right) - 3$$
-y^4 - y^2 - 3
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} - y^{2}\right) - 3$$
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
$$a = -1$$
$$b = -1$$
$$c = -3$$
Then
$$m = \frac{1}{2}$$
$$n = - \frac{11}{4}$$
So,
$$- \left(y^{2} + \frac{1}{2}\right)^{2} - \frac{11}{4}$$
Factorization [src]
/             /    /  ____\\              /    /  ____\\\ /             /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\
|    4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |    4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |      4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |      4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /||
|x + \/ 3 *sin|------------| + I*\/ 3 *cos|------------||*|x + \/ 3 *sin|------------| - I*\/ 3 *cos|------------||*|x + - \/ 3 *sin|------------| + I*\/ 3 *cos|------------||*|x + - \/ 3 *sin|------------| - I*\/ 3 *cos|------------||
\             \     2      /              \     2      // \             \     2      /              \     2      // \               \     2      /              \     2      // \               \     2      /              \     2      //
$$\left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right)$$
(((x + 3^(1/4)*sin(atan(sqrt(11))/2) + i*3^(1/4)*cos(atan(sqrt(11))/2))*(x + 3^(1/4)*sin(atan(sqrt(11))/2) - i*3^(1/4)*cos(atan(sqrt(11))/2)))*(x - 3^(1/4)*sin(atan(sqrt(11))/2) + i*3^(1/4)*cos(atan(sqrt(11))/2)))*(x - 3^(1/4)*sin(atan(sqrt(11))/2) - i*3^(1/4)*cos(atan(sqrt(11))/2))
General simplification [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
Common denominator [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
Assemble expression [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
-3.0 - y^2 - y^4
-3.0 - y^2 - y^4
Combining rational expressions [src]
      2 /      2\
-3 + y *\-1 - y /
$$y^{2} \left(- y^{2} - 1\right) - 3$$
-3 + y^2*(-1 - y^2)
Combinatorics [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
Rational denominator [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
Trigonometric part [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4
Powers [src]
      2    4
-3 - y  - y 
$$- y^{4} - y^{2} - 3$$
-3 - y^2 - y^4