Complex zeros of polynomial with no real zeros

In summary, the polynomial x^6+x^4+x^3+x^2+1 does not have any real solutions, but can be factored into (x^2-x+1)(x^4+x^3+x^2+x+1). To factor the 4th degree polynomial, one can use lateral thinking and realize that it is a factor of x^5+x^4+x^3+x^2+x. By subtracting something from this expression, the four roots can be obtained.
  • #1
bobus
4
0
I have to find the complex zeros of the following polynomial:
x^6+x^4+x^3+x^2+1
This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with:
(x^2-x+1)(x^4+x^3+x^2+x+1)

How can I factor the 4th degree polynomial now? Or how can I factor the original polynomial from beginning?

Thanks!
 
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  • #2
welcome to pf!

hi bobus! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
bobus said:
This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with:
(x^2-x+1)(x^4+x^3+x^2+x+1)

How can I factor the 4th degree polynomial now?

that was very good factoring! :biggrin:

ok, you have x4 + x3 + x2 + x + 1 …

hmm :confused: … let's do a bit of lateral thinking :rolleyes: …

what is it a factor of ? :wink:​
 
  • #3
Ok I'm pretty sure that not even X^4+x^3+x^2+x+1 has any real solution. So I guess there are 4 more complex solutions. So I was thinking about (ax^2+bx+c)(dx^2+ex+f) and trying to come up with some values for a, b, c, d, e and f but it seems to don't work...
Any idea??
 
  • #4
no, you're looking down, look up! :rolleyes:

what fifth-order expression is it a factor of? :smile:
 
  • #5
x^5+x^4+x^3+x^2+x
How does this help me?
 
  • #6
now subtract something :wink:
 
  • #7
hmmm? I guess I'm a little confused :(
 
  • #8
x^4 + x^3 +x^2 + x + 1 = (x^5 - 1)/(x - 1) (Geometric sum formula.)

You should be able to get the four roots you need.
 

FAQ: Complex zeros of polynomial with no real zeros

1. What are complex zeros of a polynomial?

Complex zeros of a polynomial are solutions to the equation where the polynomial is equal to zero. These solutions are complex numbers, meaning they have both a real and imaginary component.

2. Can a polynomial have complex zeros and no real zeros?

Yes, it is possible for a polynomial to have complex zeros and no real zeros. This means that all of the solutions to the equation are complex numbers and there are no solutions that are purely real numbers.

3. How do you find the complex zeros of a polynomial?

To find the complex zeros of a polynomial, you can use the quadratic formula or the cubic formula, depending on the degree of the polynomial. Alternatively, you can use the rational root theorem and synthetic division to determine the possible rational and irrational zeros, and then use the quadratic or cubic formula to find the remaining complex zeros.

4. What is the significance of complex zeros of a polynomial?

The complex zeros of a polynomial are important because they provide information about the behavior of the polynomial. They can tell us about the number of turning points and the end behavior of the graph of the polynomial. Complex zeros can also be used to factor a polynomial into linear and quadratic factors.

5. How do complex zeros of a polynomial affect the graph of the polynomial?

The complex zeros of a polynomial can affect the graph by creating turning points or intercepts where the graph crosses the x-axis. They can also determine the overall shape of the graph and whether it is concave up or concave down. Additionally, complex zeros can influence the symmetry of the graph if they occur in conjugate pairs.

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