How does one solve a cubic equation?

Question

anonymous · Answer

IE. $$x ^{3}-2x^{2}+1$$

mathslover · Answer

@nathan_czh First of all Welcome to OpenStudy .

mathslover · Answer

Let me know that : do you know factor theorem or factorization ?

anonymous · Answer

I only know how to factorize a quadratic equation.

mathslover · Answer

Ok wait!

mathslover · Answer

Would you like to do this by a new thing for you? Factor theorem ?

anonymous · Answer

Alright!

mathslover · Answer

@nathan_czh ?

anonymous · Answer

Yes?

mathslover · Answer

First of all, let me know the factors of the constant here... can u tell me?

anonymous · Answer

Sorry?

mathslover · Answer

What is the constant here?

anonymous · Answer

1?

mathslover · Answer

Now what are the factors of 1 ?

anonymous · Answer

1

anonymous · Answer

only

mathslover · Answer

Now put x = 1 and solve what you get? is it zero?

anonymous · Answer

yes

mathslover · Answer

So we get : x = 1 is the solution for the given equation ? right?

anonymous · Answer

yes

anonymous · Answer

But the requested answers are -3 and 2. How so?

anonymous · Answer

The original question is:
Solve $$e^{4-x}=e^2*e^{x^{2-4}}$$

mathslover · Answer

Wait! so x- 1 = 0 .. 

Now we have : (x-1) ( y)  = x^3-2x^2+1 

y = ( x^3 - 2x^2 + 1 ) / (x-1 )

mathslover · Answer

Note: A cubic equation has 3 solutions. So first solution is 3 .. .We will soon arrive to the required two more solutions. 

solve it : y = $\large{\frac{x^3-2x^2+1}{x-1}}$

anonymous · Answer

Thanks for your help. I think I can take it from here. I understand that I will have to use synthetic division to solve the equation( although I don't know how to do synthetic division) . Thank you for all your help!

mathslover · Answer

Welcome, you might like this : https://www.purplemath.com/modules/factrthm.htm