Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$-2,3 i,-3 i$$

Step-by-Step Solution

Answer

The polynomial with real coefficients that have the given zeros \( -2, 3i, -3i \) is \( f(x) = x^3 + 2x^2 - 9x - 18 \).

Step 1: Combine the complex zeros

For the complex roots, \( 3i \) and \( -3i \), they can be combined into a quadratic binomial. This is done by multiplying the factors, \( (x-3i) \) and \( (x+3i) \) together, resulting in \( x^2 + (3i)^2 = x^2 - 9 \). This yields a quadratic binomial with real coefficients.

Step 2: Combine all the zeros to create the polynomial

Combine the quadratic binomial, \( x^2 - 9 \), with the other zero, \( -2 \). This is done by multiplying \( x^2 - 9 \) with \( x + 2 \), thus giving \( f(x) = (x + 2) (x^2 - 9) \).

Step 3: Expand the polynomial

Expand \( f(x) = (x + 2) (x^2 - 9) \) to have the polynomial in standard form. By expanding, you will get \( f(x) = x^3 + 2x^2 - 9x - 18 \).

Key Concepts

Real Coefficients

When dealing with polynomials, the term "real coefficients" refers to the numbers that are placed in front of the variables, such as the numbers in equations like \( ax^2 + bx + c \). These coefficients are real numbers, which means they can be any number on the infinite number line that includes both rational numbers (like fractions) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).
Polynomials with real coefficients are crucial because they ensure the polynomial behaves predictably on a graph. When creating a polynomial from complex zeros, ensuring that all coefficients remain real contributes to the overall stability and comprehensibility of the function.

• Ensuring real coefficients involves combining complex zeros intelligently, often by pairing them up.
• This results in quadratic binomials that when multiplied with each other, eliminate the imaginary numbers, leaving you with a real-number polynomial.

With real coefficients, your polynomial can easily represent a wide variety of functions and is applicable to real-world problems.

Complex Zeros

Complex zeros come into play when dealing with polynomials involving imaginary numbers. These zeros often appear as conjugate pairs like \(3i\) and \(-3i\).
Understanding complex zeros can feel challenging at first, but they play a pivotal role in polynomials with real coefficients.

• Complex zeros occur in pairs because when dealing with real coefficients, single imaginary roots can't balance the equation.
• The conjugate pairs have the form \((a + bi)\) and \((a - bi)\), where \(b\) is the imaginary part.

By multiplying these conjugate factors together, the imaginary parts vanish, resulting in real coefficients. This operation is fundamental in forming quadratic binomials that simplify polynomials and enable further calculations without complex numbers.

Quadratic Binomial

A quadratic binomial is a polynomial expression of the form \(ax^2 + c\), which does not involve the linear coefficient \(bx\). Quadratic binomials play an important role in processing complex zeros in polynomials.
In the exercise, when we combined the complex zero factors \((x - 3i)\) and \((x + 3i)\), their product formed the quadratic binomial \(x^2 - 9\).

• The elimination of the imaginary component arises from multiplying conjugate pairs, effectively neutralizing the imaginary parts.
• This step turns the complex numbers into a real-valued expression, transforming the polynomial into one with real coefficients.

Using quadratic binomials ensures the polynomial maintains simplicity while systematically handling the complexity of imaginary numbers.