Solving x^4 + 2x^2 + x + 2 = 0 - Help Needed!

  • Thread starter Caldus
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In summary, the conversation discusses a problem with the equation x^4 + 2x^2 + x + 2 = 0. The person is seeking help in finding the solution or solutions as they are having trouble factoring the equation. It is mentioned that there are no real zeros, but it is possible to factor it over the integers. The conversation also includes a hint and a suggestion to consider the degrees of polynomials Q and R when factoring a polynomial of degree 4. There is also a question about proving that there are no zeros, to which the response is that it will become clear once the equation is factored.
  • #1
Caldus
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OK, problem with this equation:

x^4 + 2x^2 + x + 2 = 0

What is/are the solution(s)? I can't figure out how to factor this so that I can find the zeros or whatever...
 
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  • #2
Hint: It has no Real zeros.
 
  • #3
it is possible to factor it over the integers.

suppose you have a polynomial P of degree p. you want to factor it as P = Q R where Q and R are polynomials. consider the different degrees Q and R might have if p = 4 as in your case. this consideration will direct the trials and errors when looking for Q and R.
 
  • #4
How can I prove that I have no zeros?
 
  • #5
that'll be clear once you factor it.
 

1. What is the best method for solving this equation?

The best method for solving this equation depends on the complexity of the equation and your familiarity with different techniques. Some common methods for solving equations include factoring, completing the square, and using the quadratic formula.

2. How do I know if my solution is correct?

To check if your solution is correct, you can substitute the value of x into the original equation and see if it satisfies the equation. If the value of x makes the equation equal to 0, then your solution is correct.

3. Can this equation be solved without using complex numbers?

Yes, this equation can be solved without using complex numbers if all the coefficients are real numbers. However, if the coefficients are complex numbers, then the solution may involve complex numbers.

4. Is there a shortcut or trick for solving this equation?

There is no shortcut or trick for solving this equation. However, if the equation is a perfect square or can be factored easily, it may make the solving process quicker.

5. Can this equation have more than one solution?

Yes, this equation can have more than one solution. The number of solutions will depend on the degree of the equation and the nature of the equation (e.g. quadratic, cubic, etc.).

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