Mister Exam
Factor polynomial x^4-3*x^2+4
The solution
Factorization
[src]
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$$\left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right)$$
(((x + sqrt(2)*cos(atan(sqrt(7)/3)/2) + i*sqrt(2)*sin(atan(sqrt(7)/3)/2))*(x + sqrt(2)*cos(atan(sqrt(7)/3)/2) - i*sqrt(2)*sin(atan(sqrt(7)/3)/2)))*(x - sqrt(2)*cos(atan(sqrt(7)/3)/2) + i*sqrt(2)*sin(atan(sqrt(7)/3)/2)))*(x - sqrt(2)*cos(atan(sqrt(7)/3)/2) - i*sqrt(2)*sin(atan(sqrt(7)/3)/2))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{4} - 3 x^{2}\right) + 4$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -3$$
$$c = 4$$
Then
$$m = - \frac{3}{2}$$
$$n = \frac{7}{4}$$
So,
$$\left(x^{2} - \frac{3}{2}\right)^{2} + \frac{7}{4}$$
$$\left(x^{4} - 3 x^{2}\right) + 4$$
To do this, let's use the formula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -3$$
$$c = 4$$
Then
$$m = - \frac{3}{2}$$
$$n = \frac{7}{4}$$
So,
$$\left(x^{2} - \frac{3}{2}\right)^{2} + \frac{7}{4}$$
Combining rational expressions
[src]
2 / 2\ 4 + x *\-3 + x /
$$x^{2} \left(x^{2} - 3\right) + 4$$
4 + x^2*(-3 + x^2)