Problem 30 - 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 $$ f(x)=x^{3}+4 x^{2}+4 x $$

Step-by-Step Solution

Answer

The zeros of the polynomial function \(x^{3}+4 x^{2}+4 x\) are \(x = 0\) with a multiplicity of 1 (graph crosses the axis at this point) and \(x = -2\) with a multiplicity of 2 (graph touches the axis and turns around at this point).

Step 1: Understand the function

We are given the function \(f(x)=x^{3}+4 x^{2}+4 x\). This is a cubic function with real coefficients.

Step 2: Find the zeros of the function

To find the zeros of the function, we set \(f(x) = 0\) and solve for \(x\). So, we have \(x^{3}+4 x^{2}+4 x = 0\). Notice that we can factor out an \(x\) from each term, so the factored form of the equation is \(x * (x^{2} + 4x + 4) = 0\). This gives \(x = 0\) and \(x^{2} + 4x + 4 = 0\). The quadratic can be further factored to \((x + 2)^2 = 0\). So we have another root at \(x = -2\).

Step 3: Determine multiplicities and behaviour at zeros

We have two zeros: \(x = 0\) and \(x = -2\). The multiplicity of \(x = 0\) is 1 because it occurs once in the equation. For \(x = -2\), it's a double root (i.e., it occurs twice) with a multiplicity of 2. When the multiplicity is odd, as in case of \(x = 0\), the graph crosses the x axis at the corresponding x intercept. When the multiplicity is even, as in \(x = -2\), the graph touches the x-axis and turns around at the intercept.

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression consisting of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The function provided in the exercise is a cubic polynomial, which means its highest degree is three. Such functions can have varied shapes, crossing the x-axis in different ways depending on the roots and their multiplicities.

The general form of a polynomial is given by:

• \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]

where:\

• \(a_n, a_{n-1}, \ldots, a_0\) are constants (coefficients)
• \(x\) is an unknown variable
• \(n\) is a non-negative integer, representing the degree of the polynomial

The roots, or zeros, of the polynomial are the values of \(x\) for which \(f(x) = 0\). Identifying these roots helps us understand the function's behavior and graph.

Multiplicity of Zeros

The multiplicity of a zero of a polynomial function is the number of times that zero appears as a root. It relates to how the graph behaves at, or near, the intercept with the x-axis.

• Odd Multiplicity: If a zero has an odd multiplicity (like 1, 3, 5, ...), the graph of the polynomial crosses the x-axis at that zero. For example, in our solution, \(x = 0\) has a multiplicity of 1, meaning the graph will cross the x-axis at this point.
• Even Multiplicity: If a zero has an even multiplicity (like 2, 4, 6, ...), the graph merely touches the x-axis at that zero and turns back in the direction it came. For \(x = -2\), with a multiplicity of 2, the graph touches the x-axis and turns around.

Understanding multiplicity is important in sketching the graph of a polynomial and predicting its shape and intersections with the x-axis.

Graph Behavior at Intercepts

The behavior of a polynomial graph at its intercepts can give a lot of information about the function. The nature of the intercepts—whether they cross the x-axis or just touch it—depends on the multiplicity of the zeros.

• Crossing the X-Axis: As mentioned, when the multiplicity is odd, the graph crosses the x-axis. For instance, for \(x = 0\) in the example, the zero’s odd multiplicity means the graph crosses the x-axis.
• Touching the X-Axis and Turning: If the zero's multiplicity is even, the graph will touch but not cross the x-axis, instead turning back in the opposite direction. This happens at \(x = -2\), showcasing the distinct turn due to the double root.

Studying a polynomial's intercepts and understanding their behavior assists in plotting accurate graphs and answering questions about the function's real-world applications. Knowing these concepts deepens comprehension of how different values impact the graph's path and final output.