Irrational Roots
If
P(x) = x4 - 2x3 - x2 - 9x + 2 has only irrational roots, of which their real roots are between -1 and 3.
How does one find x?
-- abdeali kothari
P(x) = x4 - 2x3 - x2 - 9x + 2 has only irrational roots, of which their real roots are between -1 and 3.
How does one find x?
-- abdeali kothari
Labels: mathschallenge, SharedPuzzle
posted by Rajesh Lal at
6:53 AM
9 Comments:
One simply looks in the text book in which this was taken from. Either that or one simply asks the professor who is teaching the class with the math puzzle on the board similar to that from the movie "Good Will Hunting" where you get extra credit or perhaps get to work for the professor and copies down the answer to post on here. Then one goes to find actual riddles to post and not equations that take a ream of paper and a scientific calculator.
You don't "find" x.. that's a function defination.. you plug in any X and solve for P(x) using that forumla, then graph it..
X | P(x)
-------------
-2 | 48
-1 | 21
0 | 2
1 | -9
2 | -18
3 | -7
4 | 78
5 | 307
yeah if you wanted people to find x set the equation equal to y not P(x)
x≈{-0.6858068908914+1.57328763033i,-0.6858068908914-1.57328763033i,0.21510705634606,3.1565067252927}
Yes but there should be a way to solve this equation without graph, and without just randomly substituting.
I mean a quadratic equation can be solved by formulas rather than substituting.
Even x cube can be solved with a particular method which does not involve guess work.
I would like an answer where it does not involve guess work and by a method the answer can be got in one go.
A Solution sent by Harrell
Harrell could you PLEASE explain that!
We can get a good approximation by using newtons method.
xr(root)= x-(p(x)/p'(x))
Meaning that the root is approximately x (which in our case will be -1 and 3) - the value you get when plugging in for x divided by its derivative.
I will do this for 3 and leave -1 to the rest of the minds out there.
First plug in 3 for p(x) (3)^4-2(3^3)-3^2-9(3)+2 = -7
Then find p'(x) and since this is a polynomial that will be easy enough
p'(x)= 4x^3-6x^2-2x-9
and plug 3 in again and you should come up with 39
now our root is (approximately) 3-(-7/39)
or 3.179
For -1 use the same method just plug the -1 in everywhere I plugged in a 3
Good riddle btw Abdeali, its been a while since I've seen a tough one on ToM, a pleasant change.
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