Problem 72 - Algebra and Trigonometry

Question

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. (a) Find the polynomial with real coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\) (b) Find the polynomial with complex coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\)

Step-by-Step Solution

Answer

(a) \(x^4 - 2x^3 + 3x^2 - 2x + 2\); (b) \(x^2 - (1 + i)x + (i - 1)\)."

Step 1: Understanding the Problem

To find the polynomials, we begin by acknowledging that the roots of a polynomial and the polynomial's degree are closely connected. If a polynomial has real coefficients, complex conjugate roots occur in pairs. For complex coefficients, there is no such requirement.

Step 2: Finding Polynomial with Real Coefficients

Given roots are \(i\) and \(1+i\). Include the conjugate of \(i\), which is \(-i\), since real coefficients require conjugate pairs. Create factors: \((x-i)(x+i)(x-(1+i))(x-(1-i))\).

Step 3: Expanding for Real Coefficients

Multiply conjugate pairs: \((x-i)(x+i) = x^2 + 1\) and \((x-(1+i))(x-(1-i)) = (x-1)^2 + 1 = x^2 - 2x + 2\). Overall polynomial is: \((x^2 + 1)(x^2 - 2x + 2)\).

Step 4: Final Polynomial with Real Coefficients

Multiply the expanded factors: \((x^2 + 1)(x^2 - 2x + 2) = x^4 - 2x^3 + 3x^2 - 2x + 2\). This polynomial satisfies having real coefficients and the smallest degree with the leading coefficient as 1.

Step 5: Finding Polynomial with Complex Coefficients

For complex coefficients, use the given roots directly: \((x-i) (x-(1+i)) = (x-i)(x-1-i)\).

Step 6: Expanding for Complex Coefficients

Expand using distributive property: \((x-i)(x-1-i) = x^2 - x - ix + i - ix + i + i^2 = x^2 - x - 2ix + (i - 1)\). Simplify to get: \(x^2 - (1 + i)x + (i - 1)\).

Step 7: Final Polynomial with Complex Coefficients

The polynomial \(x^2 - (1 + i)x + (i - 1)\) has a degree of 2, has complex coefficients, and the leading term has a coefficient of 1.

Key Concepts

Real Coefficients

Polynomials with real coefficients are expressions where all the coefficients are real numbers. This means that each numerical factor in the polynomial does not have an imaginary component. One important property of such polynomials is how they handle complex roots. If a polynomial has real coefficients and includes any complex roots, these roots must occur in conjugate pairs. For example, if a polynomial has a root at \(i\), it must also have a root at its conjugate \(-i\). This ensures that all coefficients of the polynomial remain real after multiplication and expansion. Making sure complex roots appear as conjugate pairs is crucial for maintaining real coefficients in the polynomial.

Complex Conjugate Roots

Complex conjugate roots are pairs of complex numbers that have the same real part and opposite imaginary parts. For example, \(i\) and \(-i\) are conjugates because they both have a real part of 0 but their imaginary parts are +1 and -1, respectively.

• The significance of complex conjugate roots becomes apparent when dealing with polynomials with real coefficients.
• Since polynomials require real coefficients, having a single complex root would disrupt this.
• Pairing them with their complex conjugate keeps the polynomial's coefficients real as their imaginary components cancel out in the expansion process.

Understanding this property helps simplify the process of finding a real-coefficient polynomial when complex roots are involved.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the expression. It tells us the maximum number of roots (including complex and repeated roots) the polynomial can have. For example, a polynomial of degree 4 can have up to 4 roots. Understanding the degree is important because it helps in determining the simplest form of the polynomial that includes all given roots. In our example, having roots \(i\), \(1+i\), and their conjugates means the polynomial degree with real coefficients is 4. However, when complex coefficients are allowed, the smallest degree polynomial covering the roots \(i\) and \(1+i\) is of degree 2, demonstrating how root pairing impacts the polynomial's structure.

Complex Coefficients

Polynomials with complex coefficients can include both real and imaginary numbers in their terms. An advantage of using complex coefficients is the flexibility they provide in dealing with complex roots. Unlike polynomials with only real coefficients, no conjugate pair is required to maintain complex coefficients. This allows for constructing smaller degree polynomials. For instance, in our example of finding a polynomial with roots \(i\) and \(1+i\), having complex coefficients enabled us to construct a second-degree polynomial directly:

• The expression \((x-i)(x-(1+i))\) simplifies to a polynomial without needing conjugate pairs.
• This results in the polynomial \(x^2 - (1 + i)x + (i - 1)\).

Understanding complex coefficients opens up a different approach to polynomial solutions that don't adhere to the symmetry requirement in real coefficient polynomials.