Problem 27 - College Algebra

Question

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=4(x-3)(x+6)^{3} $$

Step-by-Step Solution

Answer

The function \(f(x) = 4(x - 3)(x + 6)^3\) has two zeros: \(x = 3\) and \(x = -6\), with respective multiplicities 1 and 3. The graph of the function crosses the x-axis at both zeros.

Step 1: Find the zeros

To find the zeros of the function, set \(f(x) = 0\) and solve for \(x\). Setting \(4(x - 3)(x + 6)^3 = 0\), equating each factor to zero separately gives the solutions \(x = 3\) and \(x = -6\). These are the zeros of the function.

Step 2: Determine the multiplicity of each zero and how the graph behaves at each zero

The zero \(x = 3\) comes from the factor \(x - 3\), which appears only once. So, the multiplicity of the zero \(x = 3\) is 1. Since 1 is an odd number, the graph crosses the x-axis at \(x = 3\).\n\nThe zero \(x = -6\) comes from the factor \(x + 6\), which is cubed, so it appears three times. Thus, the multiplicity of the zero \(x = -6\) is 3. Since 3 is an odd number, the graph also crosses the x-axis at \(x = -6\).

Step 3: Summary of Zeros, Multiplicities and Behaviors

To summarize, the function \(f(x) = 4(x - 3)(x + 6)^3\) has two zeros: \(x = 3\) of multiplicity 1 and \(x = -6\) of multiplicity 3. The graph crosses the x-axis at both zeros.

Key Concepts

Multiplicity of Zeros

Multiplicity of zeros in a polynomial function is a way to indicate how many times a particular zero appears as a root of the polynomial. Essentially, it tells us the degree or count of that particular solution appearing in the polynomial equation. In mathematical terms, it is the number of times a zero is repeated as a solution of the polynomial.Let’s break it down:

• If a zero has a multiplicity of 1, it means the solution appears once, hence the graph behaves differently compared to higher multiplicities at that point.
• For a zero with a multiplicity greater than 1, it means the zero appears more than once. For example, a multiplicity of 2 indicates a repeated solution, and often results in the graph touching the x-axis at this point but not passing through it.

In the case of the function \(f(x) = 4(x - 3)(x + 6)^3\), we noticed that:

• The zero \(x = 3\) has a multiplicity of 1 because its factor \((x - 3)\) appears once.
• The zero \(x = -6\) has a multiplicity of 3 because the factor \((x + 6)\) is raised to the power of 3.

Graph Behavior at Zeros

The behavior of a polynomial graph at its zeros is crucial for understanding the structure and nature of the graph. This behavior is largely determined by the multiplicity of the zeros.Here’s how multiplicity affects the graph:

• For zeros with an odd multiplicity (like 1, 3, 5, etc.), the graph will cross the x-axis. This indicates a change in the sign of the function values as it passes through the zero.
• If the multiplicity is even (like 2, 4, 6, etc.), the graph will touch the x-axis but will not cross it. The function values don’t change sign, implying a bounce back or turn around at the x-axis.

Examining the function again, \(f(x) = 4(x - 3)(x + 6)^3\):

• At \(x = 3\), the zero has an odd multiplicity of 1, causing the graph to cross the x-axis at this point.
• At \(x = -6\), the zero with an odd multiplicity of 3 also results in the graph crossing the x-axis.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. These functions succeed in modelling a vast range of phenomena both in pure mathematics and in the real-world applications.Basic Properties of Polynomial Functions:

• The degree of a polynomial is the highest power of its variable.
• The zeros or roots of the polynomial are the solutions to the equation set when the polynomial is equal to zero.
• The leading coefficient is the coefficient of the term with the highest power of the variable.

For the polynomial in our example \(f(x) = 4(x - 3)(x + 6)^3\):

• The degree is 4, since \(x + 6\) is cubed, contributing three to the degree, and \(x - 3\) contributes one, totaling four when added together.
• This degree indicates that the function can have up to four real zeros or roots, though complex zeros could also be involved.
• The leading coefficient affects the width and direction of the polynomial's graph. Here, the coefficient 4 scales the graph vertically.

Understanding polynomial functions help in graphing and predicting the behavior of these mathematical models.