Problem 25 - Algebra and Trigonometry

Question

In Exercises 25–32, 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

The nth-degree polynomial function with real coefficients satisfying the given conditions is \[ f(x) = 2(x - 1)(x^2 + 25) \] or \(f(x) = 2x^3 - 2x^2 + 50x - 50\)

Step 1: Write the polynomial as a product of binomial

First, convert the zeros into factors of the polynomial equation. The roots are 1 and \(5i\). Since \(5i\) is a root and the coefficients are real numbers, its conjugate \(-5i\) is also a root. So, the factors of the polynomial are \((x-1), (x- 5i)\), and \((x+ 5i)\).

Step 2: Construct the polynomial by multiplying the factors

We multiply the factors together to get the polynomial, \( f(x) = a (x - 1)(x - 5i)(x + 5i) \. Multiply the two complex root factors first: \( (x - 5i)(x + 5i) = x^2 + 25 \. So, \[ f(x) = a (x - 1)(x^{2}+25) \]

Step 3: Find the missing coefficient a

The coefficient \(a\) is identified by substituting \(x = -1\) and \(f(x) = -104\) into the equation derived above, and then solving for \(a\). Substituting the values into the equation, \[ -104 = a(-1 - 1)((-1)^2 + 25) \] After simplifying the above equation, we get, \[ a = -104 / -52 = 2 \]

Step 4: Write the final function

Substituting the value of \(a\) into the polynomial equation, we get our final equation as: \[ f(x) = 2(x - 1)(x^2 + 25) \] This can be further expanded if needed.

Key Concepts

Real Zeros

Real zeros are the points where a polynomial crosses or touches the x-axis on a graph. These zeros are real numbers and they make the polynomial equation equal to zero. In simpler words, a real zero of a polynomial is a solution to the equation where the polynomial value is zero. When working with polynomial functions, identifying these real zeros helps us understand the behavior of the polynomial on a graph.

For example, in the given exercise, 1 is a real zero. This means that when you input 1 into the polynomial function, you will get zero as the output. Real zeros can be easily verified by plugging them into the polynomial equation.

• They correspond to x-values of the points where the graph intersects the x-axis.
• They can be found using algebraic techniques like factoring or graphing.

Knowing real zeros helps predict the long-term behavior of the polynomial and determines end behaviors.

Complex Roots

Complex roots occur when a polynomial does not cross the x-axis at real numbered points. These are roots that include imaginary numbers, and they often come in conjugate pairs when dealing with polynomials having real coefficients. For real coefficients in a polynomial, if a complex number is a root, its conjugate must also be a root.

In our initial problem, you encounter a complex root in the form of \(5i\). Although \(5i\) is a root, the nature of real coefficients dictates that its conjugate, \(-5i\), must also be a root. This ensures that when you multiply these conjugate roots, their imaginary parts cancel out to maintain the real nature of the coefficients.

• Complex roots are always in pairs for polynomials with real coefficients.
• They do not affect the real graph intercepts directly but influence the graph's shape.

This understanding is essential in constructing polynomials accurately.

Real Coefficients

Real coefficients are numbers which multiply the variables in a polynomial, ensuring that all operations within the polynomial remain within the realm of real numbers. A polynomial with real coefficients can include complex roots, but only if they appear in conjugate pairs.

The exercise demonstrated that even though one of the roots was complex (\(5i\)), we could still maintain real coefficients by ensuring the conjugate root \(-5i\) was included. This keeps the polynomial's terms real when expanded.

• Real coefficients guarantee a polynomial translates well on a real-number coordinate graph.
• They allow the polynomial to handle both real and complex roots properly.

Understanding real coefficients allows one to construct valid and robust polynomial equations.

Nth-degree Polynomial

An \(n^\text{th}\)-degree polynomial is a polynomial equation where the highest power of the variable (usually \(x\)) is \(n\). This determines the polynomial's complexity and the maximum number of roots it can have. The degree reflects both the number of zeroes (or roots) a polynomial can have and helps analyze the behavior and shape of its graph.

In the exercise, you are tasked with constructing a cubic polynomial function. A cubic polynomial is an example of a 3rd-degree polynomial, meaning it can have up to three roots. Out of those roots:

• Some can be real, like the example's 1.
• Some can be complex, like \(5i\) and its conjugate.

The \(n^\text{th}\)-degree tells you about the function's end behavior and general shape, and it is crucial for understanding the full polynomial graph.