Problem 68 - Algebra and Trigonometry

Question

\(66-68=\) 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) The polynomial is \(x^4 - 2x^3 + 3x^2 - 2x + 2\). (b) The polynomial is \(x^2 - (1+i)x + (1+i)\).

Step 1: Understanding the Problem

We need to find two different polynomials. For part (a), the polynomial must have real coefficients and is of the smallest degree possible, with \(i\) and \(1+i\) as zeros. For part (b), the polynomial can have complex coefficients, is of the smallest degree possible, and also has \(i\) and \(1+i\) as zeros.

Step 2: Identifying Conjugate Pairs for Part (a)

For polynomials with real coefficients, complex roots appear in conjugate pairs. The given root \(i\) implies \(-i\) is also a root. The root \(1+i\) implies \(1-i\) is also a root.

Step 3: Constructing the Polynomial for Part (a)

With roots \(i, -i, 1+i,\) and \(1-i\), we can write the polynomial as \((x-i)(x+i)(x-(1+i))(x-(1-i))\). Note that \((x-i)(x+i) = x^2 + 1\) and \((x-(1+i))(x-(1-i)) = (x-1)^2 + 1\). Multiply these two results to find the polynomial of minimal degree:

Step 4: Multiply and Simplify for Part (a)

Calculate: \[(x^2 + 1)((x-1)^2 + 1) = (x^2 + 1)(x^2 - 2x + 2)\]Expand and simplify:\[= (x^2 + 1)(x^2 - 2x + 2) = x^4 - 2x^3 + 3x^2 - 2x + 2\]

Step 5: Construct the Polynomial for Part (b)

For part (b), we only need the actual zeros and do not need conjugate pairs. The polynomial can be written directly as \((x - i)(x - (1+i))\).

Step 6: Multiply and Simplify for Part (b)

Expand \((x - i)(x - (1+i))\):\( = (x - i)(x - 1 - i)\)Distribute to get:\( = x^2 - x(1+i) - i(x - 1) - i^2 = x^2 - (1+i)x - ix + i + 1 = x^2 - ix - x + i + 1 \)\(= x^2 - (1+i)x + (1+i)\).

Key Concepts

Real Coefficients

In mathematics, the term "real coefficients" refers to the numbers in a polynomial that are real numbers. Real numbers are those that you can find on the number line, including whole numbers, fractions, and decimals. These coefficients are important because they determine the real polynomial's behavior and characteristics.

In the context of our problem, when a polynomial has real coefficients, any complex roots (which are not real) must come in conjugate pairs. This means if a polynomial with real coefficients has a root of the form \(a + bi\) (where \(i\) is the imaginary unit), it must also have a root of \(a - bi\).

• Helps ensure the polynomial remains real.
• Keeps balance in the complex plane.

Knowing this is particularly useful when trying to find polynomials of the smallest degree that maintain real coefficients, as shown in the solution for part (a) of our exercise.

Imaginary Coefficients

Imaginary coefficients introduce an interesting twist when dealing with polynomials. An imaginary coefficient contains \(i\), the imaginary unit, which is defined by the property \(i^2 = -1\). Imaginary numbers extend real numbers to form complex numbers.

Polynomials with imaginary coefficients are more flexible than those with strictly real coefficients.

• These polynomials can sometimes directly use zeros from the complex plane without needing them in conjugate pairs.
• This flexibility allows for simpler polynomial forms, particularly when crafting them to a minimal degree.

In exercise part (b), we can directly use zeros like \(i\) and \(1+i\), allowing for constructs without needing conjugate pairs, resulting in polynomials that are potentially simpler and yet fit the criteria set.

Complex Numbers

Complex numbers are a combination of real and imaginary parts, expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. These numbers are pivotal in expanding the types of roots that a polynomial can have.

Understanding complex numbers is essential for grasping how polynomials can behave in higher-dimensional spaces than just real numbers.

• They allow for more flexibility in defining zeros of polynomials.
• Enhance mathematical models in various fields.

Recognizing that polynomials can have complex numbers as solutions broadens the horizon of potential polynomial functions, leading to solutions like those found in this exercise.

Polynomial Degree

The degree of a polynomial is defined as the highest power of the variable \(x\) present in the polynomial, with a non-zero coefficient specified. It's a crucial factor in determining the polynomial's shape and complexity.

The degree tells us several things:

• The maximum number of solutions or roots a polynomial can have.
• The general form of the graph of the polynomial.

In both parts of this exercise, finding polynomials of the smallest degree means finding the simplest polynomial that satisfies the given conditions.

A smaller degree often indicates a simpler polynomial, both in terms of graph shape and algebraic complexity. This pursuit of minimal degree guarantees the solution is both efficient and effective, perfectly balancing the mathematical requirements presented.