Possible for polynomial degree3 has no real zeroes? Give an example if exist or reason if not...?

Question

Ok... the real question is like this,

Is it possible to find a polnomial degree 3 that has no real zeroes? Give an example if such polynomial exist or explain why is it impossible to to find such a polynomial....

So.. the question sounds like that... one of my friend told me that there is no such thing but unable to answer / proof as he just show sketches of the graph but im not too sure.......

Can anyone clear the fog within my brain? Thank you....

Answer 1

It is possible for a polynomial of degree 3 to have no real zeroes but in that case the coefficient of the polynomial will have to be complex.

for example

(x+i)(x+2i)(x+1-i) = 0 has 3 complex roots

however if coefficients are real then it shall have at least one real root.

this follows from link below

Answer 2

any polynomial of degree 3 and in general any polynomial of odd degree has one real root.
Think about the graph represented by the polynomial function. Suppose the polynomial has positive leading coefficient. When x becomes very big (goes to infinity) then the function goes to infinity. When x takes very big NEGATIVE values ( goes to -infinity) then the function goes to -infinity too. That means that at some point the graph will intersect the x-axis, which happens in the points where the function has real roots. Hence it must have at least one root.

Answer 3

x^3 - x^2 +x + 1=0

Answer 4

If the coefficients of the polynom of third degree are real numbers you can prove that the complex roots appear by couples.That means that a+bi is root also a-bi is root.So this polynom can't have three complex roots.So there is always at least one real root