Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(5 .\) Double zero at \(x=1\), and triple zero at \(x=3\). Passes through the point (2,15) .

Step-by-Step Solution

Answer

The polynomial equation is \(f(x) = -15(x-1)^2(x-3)^3\).

Step 1: Understand the structure of the polynomial

Since the polynomial is of degree 5, and has a double zero at \(x=1\) and a triple zero at \(x=3\), we know the form of the polynomial will be \(f(x) = a(x-1)^2(x-3)^3\).

Step 2: Use the point to find the coefficient

The polynomial passes through the point \((2,15)\). Substitute \(x=2\) and \(f(x)=15\) into \(f(x) = a(x-1)^2(x-3)^3\) to find \(a\):\[ 15 = a(2-1)^2(2-3)^3 \]\[ 15 = a(1)(-1)^3 \]\[ 15 = -a \]\[ a = -15 \]

Step 3: Write the complete polynomial equation

Substitute the value of \(a\) back into the polynomial equation: \(f(x) = -15(x-1)^2(x-3)^3\).

Key Concepts

Degree of Polynomial

The degree of a polynomial is a way to describe the highest power of the variable in the polynomial equation. It is an important concept as it indicates the number of roots or zeroes that the polynomial can have. For example, if the degree is 5, it suggests the polynomial can potentially have up to 5 roots. Each root represents a value of the variable which makes the polynomial equal to zero.

• The degree gives insight into the polynomial's behavior and shape; a higher degree can mean more turns or curves in its graph.
• In the given exercise, the polynomial has a degree of 5. This stems from the arrangement of the double zero at \(x=1\) and the triple zero at \(x=3\), providing a total of 5, matching the degree.

Understanding the degree is crucial for graphing and writing polynomial equations, as it helps structure the function correctly while ensuring all aspects, like zeroes, are accounted for.

Zeroes of Polynomial

Zeroes of a polynomial, also known as roots, are the values of the variable that result in the polynomial equating to zero. These points are where the graph of the polynomial touches or crosses the x-axis. They are significant because knowing them helps determine the factors of the polynomial.
Zeroes come in different multiplicities, indicating how many times a particular root appears:

• A double zero means that this root repeats twice in the equation, evident from the factor \((x-1)^2\).
• A triple zero implies that the root appears three times, as seen in \((x-3)^3\).
• The concept of multiplicity affects the graph's behavior at these points, like touching the x-axis and turning around.

For the polynomial given in the exercise, the double zero at \(x=1\) and the triple zero at \(x=3\) are factors contributing to its degree of 5. Calculating the zeroes and understanding their multiplicity is a key step in constructing the polynomial equation.

Point on Polynomial Graph

A specific point on a polynomial graph provides valuable insight into the polynomial function. Such a point, like \((2,15)\) in the exercise, is used to calculate coefficients within the polynomial equation. This point tells us that when \(x=2\), the y-value is \(15\), which is a crucial piece of information.
Using this point helps to determine the unknown coefficient \(a\) in the polynomial equation \(f(x) = a(x-1)^2(x-3)^3\).

• Substituting \(x=2\) and \(f(x)=15\) into the equation allows us to solve for \(a\).
• This method of using a known point on the graph ensures the polynomial fits the desired graph shape perfectly.

Identifying and utilizing a specific point aids in solving polynomial equations by anchoring the graph precisely where it should pass through, ensuring accuracy in the polynomial representation. By finding the value of \(a\), we refine the equation to properly describe the entire polynomial.