Problem 6 - Applied Partial Differential Equations

Question

Prove that if \(x\) is a root of a sixth-order polynomial with real coefficients, then \(x\) is also a root.

Step-by-Step Solution

Answer

Any root \(x\) of a sixth-order polynomial with real coefficients will have its complex conjugate \(x\) also as a root if the root \(x\) is a complex number. This is according to the Complex Conjugate Root Theorem.

Step 1 Understand the Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and if a complex number is a root, then its complex conjugate must also be a root. This means that if \(x\) is complex and a root of the polynomial, then \(x\) must also be a root.

Step 2 Apply the theorem

The equation given is a sixth-order polynomial with real coefficients, therefore, it falls under the jurisdiction of the Complex Conjugate Root Theorem. With \(x\) being a root, \(x\) also becomes a root due to the theorem.

Step 3 Conclude the proof

Therefore, any root \(x\) of a sixth-order polynomial with real coefficients will have its conjugate \(x\) also as a root. Which shows that a polynomial of even order has pairs of complex conjugate roots. This also proves that an odd order polynomial has at least one real root.

Key Concepts

Polynomial with Real Coefficients

Polynomials are mathematical expressions involving sums of powers of a variable. When we say a polynomial has real coefficients, it means all the numbers multiplying the variable terms, as well as the constant term, are real numbers. Why does it matter? Because real coefficients influence the behavior of the polynomial's roots.

• If a polynomial has real coefficients and a complex number is a root, then its complex conjugate must also be a root.
• This is due to the Complex Conjugate Root Theorem, which we'll discuss more in other sections.
• The roots provide insight into the solutions of the polynomial equation.

Understanding that coefficients are real frames the bounds within which complex and real roots can exist. This characteristic is key when solving polynomial equations in complex numbers.

Complex Roots

Complex numbers are numbers in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined as \(i^2 = -1\). The complex conjugate, a related concept, is \(a - bi\). Complex roots come in pairs, and if a polynomial has real coefficients, its complex roots must come in conjugate pairs.

• For example, if \(3 + 2i\) is a root, then \(3 - 2i\) will also be a root.
• This characteristic helps ensure that the polynomial's calculation results in real coefficients.
• It also prevents isolated complex occurrences in the polynomial's solutions.

Being aware of complex roots' behaviors allows us to anticipate the solutions when dealing with polynomials with real coefficients.

Even and Odd Order Polynomials

A polynomial's order, also known as degree, is determined by the highest exponent of its variable term. The order highly influences the root characteristics and symmetry of a polynomial equation. Let's explore the difference between even and odd order polynomials.

• Even order polynomials can accommodate pairs of conjugate complex roots without affecting the realness of the overall expression.
• They often have an even number of roots, allowing for this pairing naturally.
• Odd order polynomials, on the other hand, have at least one real root because they cannot pair all complex roots without leftovers.

Each of these properties helps us predict the necessity of real roots in odd order polynomials and the structured pairing of complex roots in even order ones. This makes fundamental prediction and calculus on polynomial roots far clearer.