Root Theorem for Polynomials of Degree > 2

SummarizerIn summary, there is a theorem that states if a polynomial with a degree greater than 1 and real coefficients has a complex root, then its conjugate is also a root. This is easy to prove for degree 2 polynomials, but may be more difficult for higher degrees. While this concept is sometimes mentioned in relation to the Fundamental Theorem of Algebra, it does not have a specific name.
  • #1
hedlund
34
0
What is the theorem that states if [tex] \Omega [/tex] is a polynom with degree > 1 with real coefficients. If there exists a complex number [tex] z = a + bi [/tex] such that [tex] \Omega(a+bi)=0 [/tex] then [tex] \overline{z} = a - bi [/tex] is also a root of [tex] \Omega [/tex]? For [tex] \Omega(x) = x^2 + px + q [/tex] with p and q real then if a+bi is a root then a-bi is also a root if [tex] b \neq 0 [/tex], that one is easy but I don't think it's easy for degree > 2 to prove it that's why I'm search for it's name.
 
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  • #2
it doesn't have a name, as far as i know, and it is easy to prove. if z is a root of P, then z* is a root of P*, where * denotes conjugation, and by P*, I mean the polynomial where you replace the coeffs with their conjugates. (You understand that (uv)*=u*v*?)
 
  • #3
It does get mentioned along with FTA but i wouldn't bet on it having some special name.

-- AI
 

Related to Root Theorem for Polynomials of Degree > 2

What is the Root Theorem for Polynomials of Degree > 2?

The Root Theorem is a mathematical principle used to determine the possible number of real roots for a polynomial function of degree greater than 2. It states that the number of positive real roots is equal to the number of sign changes in the coefficients of the polynomial, while the number of negative real roots is equal to the number of sign changes in the coefficients of the polynomial when all the even powers are replaced with their absolute values.

How is the Root Theorem used in polynomial graphing?

The Root Theorem is used to determine the x-intercepts or roots of a polynomial function. By finding the number of positive and negative real roots, we can determine the number of times the graph of the polynomial will cross the x-axis. This information can help us sketch the overall shape of the graph.

What is the difference between real and complex roots in polynomials?

Real roots are values of x that make the polynomial function equal to zero, while complex roots are pairs of numbers that make the polynomial function equal to zero. Complex roots always come in pairs, and they involve the imaginary unit, i, which is equal to the square root of -1.

Can the Root Theorem be used for all polynomials?

No, the Root Theorem only applies to polynomials of degree greater than 2. For polynomials of degree 2 or less, the number of real roots can be found using the quadratic formula.

How can the Root Theorem help us with polynomial division?

The Root Theorem can be used to find potential rational roots of a polynomial. By factoring the polynomial using the potential roots, we can simplify the polynomial and make the division process easier. The Root Theorem can also help us determine if a polynomial is irreducible, meaning it cannot be factored any further.

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