Mister Exam
Factor -y^4-7*y^2-15 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- y^{4} - 7 y^{2}\right) - 15$$
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
$$a = -1$$
$$b = -7$$
$$c = -15$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{11}{4}$$
So,
$$- \left(y^{2} + \frac{7}{2}\right)^{2} - \frac{11}{4}$$
$$\left(- y^{4} - 7 y^{2}\right) - 15$$
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
$$a = -1$$
$$b = -7$$
$$c = -15$$
Then
$$m = \frac{7}{2}$$
$$n = - \frac{11}{4}$$
So,
$$- \left(y^{2} + \frac{7}{2}\right)^{2} - \frac{11}{4}$$
Factorization
[src]
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$$\left(x + \left(\sqrt[4]{15} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)} - \sqrt[4]{15} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{15} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)} + \sqrt[4]{15} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)} + \sqrt[4]{15} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)} - \sqrt[4]{15} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{11}}{7} \right)}}{2} \right)}\right)\right)$$
(((x + 15^(1/4)*sin(atan(sqrt(11)/7)/2) + i*15^(1/4)*cos(atan(sqrt(11)/7)/2))*(x + 15^(1/4)*sin(atan(sqrt(11)/7)/2) - i*15^(1/4)*cos(atan(sqrt(11)/7)/2)))*(x - 15^(1/4)*sin(atan(sqrt(11)/7)/2) + i*15^(1/4)*cos(atan(sqrt(11)/7)/2)))*(x - 15^(1/4)*sin(atan(sqrt(11)/7)/2) - i*15^(1/4)*cos(atan(sqrt(11)/7)/2))
Combining rational expressions
[src]
2 / 2\ -15 + y *\-7 - y /
$$y^{2} \left(- y^{2} - 7\right) - 15$$
-15 + y^2*(-7 - y^2)