What approach should i take to finding the values of x for this equation
$\displaystyle \frac{x^3-3x^2+1}{x^3}=0$
First multiply both sides by $\displaystyle x^3$ getting
$\displaystyle x^3-3x^2+1=0$
Then recall that the zeros of any polynomial equation of the form $\displaystyle a_nx^n+a_{n-1}x^{n-1}......a_1x+a_0=0$ can be expressed as a ratio of $\displaystyle \frac{p}{q}$ where $\displaystyle p$ are all possible factors of $\displaystyle a_0$ and $\displaystyle q$ are the possible factors of the leading coefficient $\displaystyle a_n$.
Obviously some of the zeros are of a repeating nature, so you can narrow the field down a little bit. In more complicated expressions, you would normally use synthetic division to test the roots, but here it should be evident which ones are the right choice.
This could help.
Box 5.3a: The Cubic Formula
© 1997 by Karl Hahn
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Solving a Cubic
Four centuries before European mathematicians solved the problems of the cubic and quartic equations, Indian mathematician and astronomer, Bhaskara, had solved these problems. You can see a biography of Bhaskara by clicking here.
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Note that this material is not at all likely to be on the exam. The formula is not even all that useful. It's given here merely to satisfy your curiosity. So you may skip this box if you like.
If you are given a cubic equation in the form of
x3 + px2 + qx + r = 0 eq. 5.3a-1
and need to solve for x, then the first thing you do is substitute variables. Everywhere you see an x in the cubic, replace it with
p
x = u - eq. 5.3a-2
3
When you get done squaring and cubing this expression, then substituting stuff back in and gathering like terms, you will get
u3 + au + b = 0 eq. 5.3a-3a
where
p2
a = q - eq. 5.3a-3b
3
and
pq 2p3
b = r - + eq. 5.3a-3c
3 27
Now compute A and B by
A =
eq. 5.3a-4a
and
B =
eq. 5.3a-4b
Then you have the following solutions for u
u = A + B eq. 5.3a-5a
____
u = -(1/2)(A + B) + √-3/4 (A - B) eq. 5.3a-5b
____
u = -(1/2)(A + B) - √-3/4 (A - B) eq. 5.3a-5c
Of course, you know how convert from the u solutions to the x solutions (hint: look at 5.3a-2).
You may be troubled by the expression, √-3/4, which is not a real number, but is a complex number. Note that A and B will also nonreal complex numbers whenever
b2 a3
+ < 0 eq. 5.3a-6
4 27
So the cubic formula does require that you understand the arithmetic of complex numbers.
And if you are not up on complex numbers, don't worry. We will be reviewing them in a later section.