Problem 72 - Precalculus Mathematics for Calculus

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) Polynomial: \(x^4 - 2x^3 + 3x^2 - 2x + 2\); (b) Polynomial: \(x^2 - 2x + (i + 1)\).

Step 1: Identify Zeros and Conjugates

For part (a), identify that the zeros given are complex. The polynomial with real coefficients requires that complex zeros occur in conjugate pairs. Therefore, the zeros needed are \( i, -i, 1+i, \) and \( 1-i \). For part (b), use only the given zeros, \( i \) and \( 1+i \), as complex coefficients allow non-conjugate zeros.

Step 2: Form Polynomials from Zeros

For part (a), derive factors using zeros: \((x - i)(x + i)(x - (1+i))(x - (1-i))\). For part (b), use factors \((x - i)(x - (1+i))\).

Step 3: Expand the Factors for Real Coefficient Polynomial

Expand the pairs for real coefficients polynomial. First pair: \((x - i)(x + i) = x^2 + 1\). Second pair: \((x - (1+i))(x - (1-i)) = (x-1)^2 + 1 = x^2 - 2x + 2\).

Step 4: Multiply the Expanded Real Coefficient Factors

Multiply the two quadratic polynomials: \((x^2 + 1)(x^2 - 2x + 2)\). Expand this multiplication to get the resulting polynomial: \((x^2 + 1)(x^2 - 2x + 2) = x^4 - 2x^3 + 2x^2 + x^2 - 2x + 2 = x^4 - 2x^3 + 3x^2 - 2x + 2\).

Step 5: Expand the Factors for Complex Coefficient Polynomial

For part (b), multiply the factors directly because there is no need for conjugate pairs: \((x-i)(x-(1+i)) = x^2 - (1+i)x - ix + (1)(-i) \). Simplify to obtain \( x^2 - (1+i+1)x + i - i^2 = x^2 - (1+i+1)x + i + 1 = x^2 - 2x + (i + 1) \).

Step 6: Write the Polynomial Results

For part (a), the polynomial is \(x^4 - 2x^3 + 3x^2 - 2x + 2\). For part (b), the polynomial is \(x^2 - 2x + (i + 1)\).

Key Concepts

Zeros of Polynomials

The concept of zeros in polynomials is fundamental. These are the values for which the polynomial becomes zero. For instance, if a polynomial is expressed as \( f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n \), the zeros are the solutions of \( f(x) = 0 \). In the context of complex numbers, a polynomial can have both real and complex zeros. Complex numbers often come in pairs known as complex conjugates, especially when dealing with polynomials that have real coefficients. This property ensures that the product or result remains rooted in real numbers. Understanding zeros is crucial for determining the factors of a polynomial and further expanding it into a standard polynomial form.

Complex Conjugates

Complex conjugates are pairs of complex numbers that play a significant role when working with polynomials with real coefficients. If you have a complex number \( a + bi \), its conjugate is \( a - bi \). This means they only differ in the sign of the imaginary part. When included as zeros in a polynomial, conjugate pairs ensure that the polynomial remains real. Thus, a polynomial with real coefficients must include complex zeros in these conjugate pairs. For instance, if \( i \) is a zero, \( -i \) must also be included to retain real coefficients. This concept elevates the polynomial to the smallest even degree necessary to incorporate such zeros. It's a critical rule to remember when constructing polynomials with given complex numbers.

Real Coefficients Polynomial

When constructing a polynomial with real coefficients, one of the main requirements is that complex roots must appear in conjugate pairs. This is crucial because it helps ensure that all coefficients of the polynomial are real numbers. The real coefficients provide a way to express the polynomial without imaginary numbers appearing in the factors. For example, if \( i \) and \( 1+i \) are zeros, to maintain real coefficients, \( -i \) and \( 1-i \) must also be zeros. The key is ensuring the complex components cancel out in the multiplication process, leaving zero imaginary parts in the resulting polynomial expression. Real coefficient polynomials are especially useful in many practical applications, where results need to remain within the real number system.

Polynomial Degree

The degree of a polynomial is determined by the highest power of its variable. It defines the polynomial's complexity and the number of possible roots or zeros it can have. For instance, a fourth-degree polynomial could potentially have 4 zeros in the complex number system. When dealing with real coefficients, the degree must accommodate complex conjugate pairs, making the overall degree even if there’s an odd count of complex zeros. Conversely, complex coefficient polynomials, where conjugates are not mandatory, can have a degree that is solely determined by the non-conjugate zeros provided. Understanding the polynomial degree helps in anticipating the behavior of the polynomial and in simplifying its expansion into factors.