Mister Exam
Factor polynomial x+9*x^2+27*x+162
The solution
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2 x + 9*x + 27*x + 162
$$\left(27 x + \left(9 x^{2} + x\right)\right) + 162$$
x + 9*x^2 + 27*x + 162
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(27 x + \left(9 x^{2} + x\right)\right) + 162$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 9$$
$$b = 28$$
$$c = 162$$
Then
$$m = \frac{14}{9}$$
$$n = \frac{1262}{9}$$
So,
$$9 \left(x + \frac{14}{9}\right)^{2} + \frac{1262}{9}$$
$$\left(27 x + \left(9 x^{2} + x\right)\right) + 162$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 9$$
$$b = 28$$
$$c = 162$$
Then
$$m = \frac{14}{9}$$
$$n = \frac{1262}{9}$$
So,
$$9 \left(x + \frac{14}{9}\right)^{2} + \frac{1262}{9}$$
Factorization
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/ ______\ / ______\ | 14 I*\/ 1262 | | 14 I*\/ 1262 | |x + -- + ----------|*|x + -- - ----------| \ 9 9 / \ 9 9 /
$$\left(x + \left(\frac{14}{9} - \frac{\sqrt{1262} i}{9}\right)\right) \left(x + \left(\frac{14}{9} + \frac{\sqrt{1262} i}{9}\right)\right)$$
(x + 14/9 + i*sqrt(1262)/9)*(x + 14/9 - i*sqrt(1262)/9)
Combining rational expressions
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162 + x*(28 + 9*x)
$$x \left(9 x + 28\right) + 162$$
162 + x*(28 + 9*x)