Problem 31 - 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. \(n=3 ;-5\) and \(4+3 i\) are zeros; \(f(2)=91\)

Step-by-Step Solution

Answer

The polynomial function with real coefficients, degree 3, with zeros at -5, 4+3i, and 4-3i and a function value of 91 at x=2 is \(f(x) = (x+5)(x^2-8x+25)\).

Step 1: Determine the Polynomial Function (General Form)

Given the zeros, \(-5\), \(4+3i\), and \(4-3i\), the polynomial function \(f(x)\) can be written in standard form as \(f(x) = a(x+5)(x-(4-3i))(x-(4+3i))\).

Step 2: Simplify the Polynomial Function

Use the binomial theorem to simplify \(f(x) = a(x+5)(x-4+3i)(x-4-3i)\) and write it as a polynomial function with real coefficients. This simplification gives \(f(x) = a(x+5)(x^2-8x+25)\).

Step 3: Determine the Coefficient 'a'

Using the given condition that \(f(2) = 91\), substitute \(x=2\) in \(f(x)\) and solve for 'a'. So, \(91 = a*(2+5)*(4-16+25)\). Solving this equation gives \(a = 1\).

Step 4: Write the Final Polynomial Function

Substitute \(a = 1\) in \(f(x) = a(x+5)(x^2-8x+25)\), which gives the final polynomial function \(f(x) = (x+5)(x^2-8x+25)\).

Key Concepts

Zeros of a Polynomial

A critical aspect of understanding polynomials is finding their zeros, also known as roots or solutions. For an nth-degree polynomial, these zeros are the values of the variable that make the polynomial equal to zero. In the given problem, the zeros are

• -5
• 4 + 3i
• 4 - 3i

The presence of imaginary numbers implies that the zeros include complex numbers, and it's important to note that these complex zeros occur in conjugate pairs. Identifying zeros is essential since they indicate the x-intercepts of the polynomial graph, where the curve crosses the x-axis. For real zeros, these points can be visually found on a graph, while imaginary roots don't cross but still influence the graph's shape.

Complex Conjugates in Polynomials

Complex numbers, which include both real and imaginary parts, play a significant role in polynomial functions. When a polynomial has real coefficients but complex zeros, these zeros must occur in conjugate pairs to ensure that the final polynomial has only real coefficients.

In our exercise, since 4 + 3i is a zero, its complex conjugate 4 - 3i must also be a zero. When these two are multiplied, the imaginary parts cancel each other, resulting in a polynomial expression with real coefficients, which simplifies the polynomial to real number factors when solved out. This encapsulates the property of polynomials in real-number spaces where any complex roots must come in pairs.

Polynomial Graphing

Graphing a polynomial provides a visual illustration of its behavior. By examining the graph, we can observe the roots, understand the polynomial's end behavior, and analyze critical points like maxima and minima. The zeros of the polynomial indicate where the graph touches or crosses the x-axis.

In our nth-degree polynomial (specifically a 3rd-degree, or cubic, polynomial), these zeros determine how the graph behaves and transitions through the x-axis. The presence of complex roots affects the graph's directionality but won't change where the curve intersects the x-axis, which highlights only the real zeros. Graphing utilities can help verify the solutions by providing a clear depiction of these interactions and confirming the calculated function behavior aligns with intended outcomes.

Nth-Degree Polynomial

The degree of a polynomial is the highest power of the variable in its expression. It defines many critical properties of the polynomial, including the number of zeros and the polynomial's end behavior. For an nth-degree polynomial:

• The polynomial can have at most n zeros.
• Its graph can intersect the x-axis at most n times.
• The number of inflections, or changes in the concavity, will generally be n-1.

For example, in our 3rd-degree polynomial solution, we observe three zeros, which match the degree of the polynomial. Each polynomial degree correlates with specific characteristics that help determine both its structure and graph. For any positive integer n, a thorough understanding of these attributes can significantly aid in understanding the polynomial's complexities and interactions.