jake2954 said:
The equation X^4 - 3X^3 + 19X^2 +53x - 174 has four roots. If two of the roots are real and equal to x=-1 and x=3 , by a process of long division and solving a quadratic equation, find the two complex roots.
Please can you help me with this question as I am struggling to anwer it.
Thanking you in advance
Jake
Hi Jake
EDIT: ARE YOU SURE YOU WROTE BOTH THE QUARTIC AND THE ROOTS CORRECTLY?
(-1)[sup:32sqivez]4[/sup:32sqivez] - 3(-1)[sup:32sqivez]3[/sup:32sqivez] + 19(-1)[sup:32sqivez]2[/sup:32sqivez] + 53(-1) - 174 = 1 + 3 + 19 - 53 - 174 = 23 - 227 = - 204, which is not quite zero.
I give help but do not solve your problems for you. Most people here do the same.
Did you know that if P(x) is a polynomial of degree n and P(a) = 0, there exists Q(x) such that (x - a) * Q(x) = P(x) and Q(x) is a polynomial of degree (n - 1)?
This is a very important theorem based on the Fundamental Theorem of Algebra.
So what is the relevance of this theorem to your problem?
It means that x[sup:32sqivez]4[/sup:32sqivez] - 3x[sup:32sqivez]3[/sup:32sqivez] + 19x[sup:32sqivez]2[/sup:32sqivez] + 53x - 174 = (x + 1) * (x - 3) * R(x) = (x[sup:32sqivez]2[/sup:32sqivez] - 2x - 3) * R(x), where R(x) is a quadratic. Do you see why?
OK. So that means R(x) = (x[sup:32sqivez]4[/sup:32sqivez] - 3x[sup:32sqivez]3[/sup:32sqivez] + 19x[sup:32sqivez]2[/sup:32sqivez] + 53x - 174) / (x[sup:32sqivez]2[/sup:32sqivez] - 2x - 3).
Do you know how to do algebraic long division?