Problem 29 - 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. \(n=3 ; 1\) and \(5 i\) are zeros; \(f(-1)=-104\)

Step-by-Step Solution

Answer

Without the specific values provided, an exact short answer cannot be given. However, the standard form of the polynomial would roughly resemble \(f(x) = a(x - 1)(x^2 + 25)\), with \(a\) being a specific coefficient.

Step 1: Identify all the roots

The roots provided are 1 and \(5i\). According to the Conjugate Pair Theorem, which states that if a complex number is a root of a polynomial with real coefficients, then its conjugate is also a root of the function, it's clear that \(-5i\) would be another root.

Step 2: Construct the polynomial

The notation \(i\) represents a complex number. With all of the roots available, the polynomial can be expressed as a product of first-degree factors: \(f(x) = a(x - 1)(x - 5i)(x + 5i)\).

Step 3: Calculate the undetermined coefficient

To find the coefficient \(a\), use the condition that \(f(-1)=-104\). So, substitute \(x = -1\) in the function: \[-104 = a(-1 - 1)((-1) - 5i)((-1) + 5i)\]. Solve the equation for \(a\) to get the coefficient's value.

Step 4: Write the final polynomial

Substitute the determined value of \(a\) into the polynomial. The simplified form of the polynomial function will be the final polynomial.

Key Concepts

Real Coefficients

When we talk about polynomial functions, one common condition is that they have real coefficients. Real coefficients mean that all the numbers in front of the variables in our polynomial are real numbers. This is important in mathematics because it affects the roots of the polynomial.

Real numbers include all the numbers we can think of on a number line, such as:

• Whole numbers, like 1, 2, 3.
• Fractions, like 1/2, 3/4.
• Negative numbers, like -2, -5.
• Decimal numbers, like 0.5, 3.14.

When we construct a polynomial function using these coefficients, we can easily relate them to other mathematical concepts like complex roots and real roots. Understanding that real coefficients in a polynomial require complex roots to occur in conjugate pairs is key to solving these types of problems.

Complex Roots

Complex roots in polynomials often involve numbers that include the imaginary unit, denoted by the letter i. The imaginary unit is defined by the property that \[i^2 = -1.\] Using complex numbers allows us to handle solutions that do not lie on the real number line.

In a polynomial with real coefficients, the presence of a complex root necessitates that its conjugate also be a root. This holds true due to the primary feature of complex numbers where each complex number has a conjugate, formed by changing the sign of the imaginary part. For example, given the complex root \(5i\), its conjugate would be \(-5i\). Together, these roots ensure that when combined in a polynomial, the result maintains real coefficients throughout.

Conjugate Pair Theorem

The Conjugate Pair Theorem is crucial when dealing with polynomial functions with real coefficients. It states that for any polynomial with real coefficients, if it has a complex root then the conjugate of that root is also a root of the polynomial. This theorem helps ensure that any polynomial with real numbers as coefficients will indeed have only real sum, product, and combinations when expanded.

Consider a polynomial where

• one root is a complex number, say \(a + bi\),
• the conjugate root would then be \(a - bi\).

These pairs are essential for maintaining the integrity of real coefficients in a polynomial. So in our scenario, given the complex root \(5i\), its conjugate \(-5i\) must likewise be considered for accurate polynomial construction.

Graphing Utility

A graphing utility is an invaluable tool for visualizing polynomial functions and confirming their properties. Graphing utilities allow students to easily check whether the polynomial function behaves as expected. They can graph the entire function to find approximate values for zeros or to observe the behavior of the function over a specified range.

For a third-degree polynomial, like the one in our example, using a graphing utility can help:

• Identify where the function crosses the x-axis, indicating real zeros.
• Verify that the function satisfies any given condition, such as ensuring that \(f(-1) = -104\).

Visualizing the polynomial through graphing helps enhance understanding of how the theoretical aspects align with practical results. Additionally, students can use graphing utilities to explore further scenarios and test changes in real-time.