factorising

Nov 2009
7
0
hey, i'm finding it difficult how to factorise this equation:

t³ - 3t² + 4 = 0

i've found out the answer already, but i really dont understand how i would get the answer- It would be very helpful if someone could give me a step to step guide on how to get to the answer.

lolz..i've used the word answer so many times in this thread but i don't want to give it away incase some MHF users would like to solve the question themselves.(Happy)
 

masters

MHF Helper
Jan 2008
2,550
1,187
Big Stone Gap, Virginia
hey, i'm finding it difficult how to factorise this equation:

t³ - 3t² + 4 = 0

i've found out the answer already, but i really dont understand how i would get the answer- It would be very helpful if someone could give me a step to step guide on how to get to the answer.

lolz..i've used the word answer so many times in this thread but i don't want to give it away incase some MHF users would like to solve the question themselves.(Happy)
Hi ladythugette,

You know, by the Rational zeros theorem, that if it has any rational roots at all,
they will be +/- the factors of 4.

I tried 2 and found that to be a zero, and used synthetic division to find the other two.

Code:
2 | 1 -3  0  4
       2 -2 -4
   ---------------
    1 -1 -2  0
The depressed polynominal is now a quadratic: \(\displaystyle x^2-x-2\) and can easily be factored.
 
Nov 2009
7
0
i've never heard of the rationals zero theorem before.i'm trying to understand what you mean but my brain seems to working very slow today.is there an alternative method that could be used to work out this question? or if not, could you explain the rationals zero theorem to me again please.
 

skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
i've never heard of the rationals zero theorem before.i'm trying to understand what you mean but my brain seems to working very slow today.is there an alternative method that could be used to work out this question? or if not, could you explain the rationals zero theorem to me again please.
here is an explanation from the beginning ...

The Rational Roots Test
 
May 2010
1
0
hey, i'm finding it difficult how to factorise this equation:

t³ - 3t² + 4 = 0

i've found out the answer already, but i really dont understand how i would get the answer- It would be very helpful if someone could give me a step to step guide on how to get to the answer.

lolz..i've used the word answer so many times in this thread but i don't want to give it away incase some MHF users would like to solve the question themselves.(Happy)
a way you could do this is to look for similar factors for example in this case:
t³ - 3t² + 4 = 0
u could rewrite the equation as t³ + 1 - 3t² + 3 = 0
now t³ + 1 = (t + 1)(t² - t + 1) and -3t² + 3=-3(t² - 1)=-3(t - 1)(t + 1).
As you can see we have same factors so we can write the equation as
(t + 1)[t² - t + 1 - 3(t - 1)]=(t + 1)(t² - 4t + 4)=(t + 1)(t - 2)²=(t + 1)(t - 2)(t - 2)
So there's the solution.