Can All Coefficients 'a' and 'b' Be Roots in Their Own Set of Quadratics?

A and B in the quadratic equations. In summary, the conversation is about whether it is possible for all values of A and B to be the roots of 'n' quadratic equations in the form of x^2 + aix + bi, where all values of A and B are different. The person asking the question has proven that it is possible for n=2, but is having difficulty with n>3 and is seeking clarification and possible constraints. The responder suggests that there may be a problem with n>3 because there are no fixed positions for each value of A and B in the quadratic equations.
  • #1
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1. Homework Statement :
We have 'n' quadratics is the form x^2 +aix+bi
All values of A and b are different.
Is it possible to have all values of A and B as the roots of the n quadratics

2. The attempt at a solution:
Well. I know that all values of 'a' and 'b' must (if possible) be in one of the n (x-t)(x-s) where t and s are two values of a and/or b. I have proven that when n=2 its possible for all values, however i have difficulty with n>3
 
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  • #2
Could you please clarify the question? Are there any other constraints to the question (like the roots must be real/complex)?

Why should there be a problem when n>3?
 
  • #3
Well there are no other restraints. Only all the values of a and b are different. I assume that both complex and real numbers are possible.

The answer to your second question I was having difficulty with N>3 is because there are not fixed 'positions' for each value
 

What is a polynomial root?

A polynomial root is a value that, when substituted into a polynomial equation, makes the equation equal to zero. It is the solution to the equation and can be found by setting the polynomial equal to zero and solving for the variable.

How do I find the roots of a polynomial?

To find the roots of a polynomial, you can use various methods such as factoring, using the quadratic formula, or using the rational roots theorem. Each method may be more suitable for different types of polynomials, so it is important to understand the different techniques and when to use them.

Can a polynomial have more than one root?

Yes, a polynomial can have multiple roots. This depends on the degree of the polynomial and the number of times a certain value appears as a root. For example, a quadratic polynomial can have a maximum of two roots, while a cubic polynomial can have a maximum of three roots.

What is the difference between real and complex roots?

Real roots are solutions to a polynomial equation that are real numbers, meaning they can be found on the number line. Complex roots, on the other hand, involve imaginary numbers and cannot be found on the number line. They are typically expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Why are polynomial roots important?

Polynomial roots are important because they help us solve equations and understand the behavior of polynomial functions. They also have real-world applications in areas such as engineering, physics, and economics. Additionally, the number of roots can tell us important information about the polynomial, such as its degree and number of turning points.

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