Problem 34 - College Algebra

Question

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=4 ;-2,-\frac{1}{2},\) and \(i\) are zeros; \(f(1)=18\)

Step-by-Step Solution

Answer

The nth-degree polynomial function is \(P(x) = (x + 2)(2x + 1)(x^2 + 1)\).

Step 1: Arrange the Polynomial

The polynomial function can be initially arranged as asked, the result is \[P(x) = k(x + 2)(2x + 1)(x - i)(x + i)\]

Step 2: Simplify Polynomial

A simplification of the polynomial accomplished by factoring gives \[P(x) = k(x + 2)(2x + 1)(x^2 + 1)\]

Step 3: Calculate Constant k

Plug \(x=1\) into the polynomial and set it to be equal to 18 to solve for the constant k.\[18 = k(1+2)(2 \cdot 1 +1)(1^2 + 1)\]\[18 = 3k(3)(2)\]Solving for k gives \(k=1\).

Key Concepts

Real Coefficients

In polynomial functions, coefficients that are real numbers are considered real coefficients. These are numbers that have no imaginary part. In other words, they are the numbers we typically use for counting and measuring. When working with polynomial functions that have real coefficients, it's crucial because they ensure the function behaves in predictable ways over the set of real numbers. This characteristic makes real coefficients suitable for graphing and analyzing using just real numbers.
In our exercise, since the polynomial has real coefficients, any complex zeros must appear in conjugate pairs. This happens because polynomial coefficients dictate the structure and solutions of the polynomial equation. For example, if a polynomial has a zero at \(i\), another zero must exist at \(-i\) to maintain the real nature of the coefficients. Understanding real coefficients allow us to further control and predict the behavior of polynomial functions on the real number line.

Real Zeros

Real zeros of a polynomial are the values of x where the polynomial equals zero. These zeros can also be called roots or x-intercepts. In simpler terms, they are the points where the graph of the polynomial crosses the x-axis. For polynomial functions with real coefficients, finding real zeros is straightforward as it means literally finding the points on the real number plane where the polynomial evaluates to zero.
In our given problem, \(-2\) and \(-\frac{1}{2}\) are the real zeros of the polynomial. These real zeros can be directly plugged into the polynomial equation to produce a zero output. This means when \(x = -2\) or \(x = -\frac{1}{2}\), the value of the polynomial function equates to zero, showing two intersections of the graph with the x-axis.

Complex Zeros

Complex zeros, as the name suggests, include imaginary numbers and are more abstract than real zeros. Imaginary numbers are often denoted with an "i," which is the square root of -1. Complex zeros often can’t be seen directly on a graph because they do not lie on the real number plane. However, they are essential to the complete solution of polynomial functions, especially since they ensure the presence of all necessary zeros when factoring polynomials with real coefficients.
According to the exercise, \(i\) is a complex zero. Due to the nature of real coefficients, another zero must exist at \(-i\), even if it's not explicitly given in the list of conditions. Pairing zeros like \(i\) and \(-i\) keeps the coefficients real and the polynomial function balanced. The presence of these complex zeros ensures that the degree of the polynomial, as specified in the problem, is correctly observed.

Graphing Utility

Graphing utilities are wonderful tools that help visualize the behavior of polynomial functions. They are particularly helpful when analyzing functions that are too complex to solve by hand or when verifying solutions, like zeros, calculated manually. By inputting the polynomial function into a graphing utility, one can see exactly where the graph intersects with the x-axis, which corresponds to the real zeros. It also offers insight into the polynomial's shape and behavior, driven by its coefficients and degree.
In this exercise, using a graphing utility can verify whether the zeros at \(-2\), \(-\frac{1}{2}\), and the impacts of complex conjugates \(i\) and \(-i\) are graphically correct. Additionally, it ensures the computed function value of \(f(1)=18\) aligns with the visual representation on the graph, affirming the correctness of the solutions found manually. Utilizing a graphing utility fosters a deeper understanding of the interplay between algebraic solutions and their graphical representations.