Problem 29 - College Algebra

Question

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=4(x-3)(x+6)^{3}$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = 4(x-3)(x+6)^3\) are \(x=3\) with a multiplicity of 1 and \(x=-6\) with a multiplicity of 3. The graph of the function crosses the x-axis at x=3 and touches the x-axis and turns around at x=-6.

Step 1: Finding the Zeros

In order to find the zeros of the polynomial function, set \(f(x)\) equal to zero and solve the resulting equation for \(x\): \(0 = 4(x-3)(x+6)^3\). The solution to this equation will give the x-values where the function crosses or touches the x-axis, i.e., the zeros of the function. This gives \(x=3\) and \(x=-6\).

Step 2: Determining the Multiplicity

The multiplicity of a root is found out by observing the power to which it raised. For every root \(r\), if it is raised to the power of \(n\) then the multiplicity of that root is \(n\). Here, since \(x=3\) is raised to the power of 1, it has the multiplicity 1, while \(x=-6\) is raised to the power of 3, so its multiplicity is 3.

Step 3: Understanding the Graph Behavior

When the multiplicity of the root is odd, the graph of the function will cross the x-axis at that root's location; if the multiplicity is even, the graph touches the x-axis and turn around. So, the graph of function will cross the x-axis at \(x=-6\) and will touch and turn around the x-axis at \(x=3\).

Key Concepts

Multiplicity of Roots

Multiplicity is a term used to describe how many times a particular root is repeated in a polynomial function.
For instance, when we have a polynomial like \(f(x)=4(x-3)(x+6)^3\), the expression inside each parenthesis represents possible roots of the function.
The number of times these roots are repeated is known as their multiplicity.

• If the expression \(x-r\) appears in the factored form of the function with an exponent \(n\), then \((x=r)\) is a root with multiplicity \(n\).
• This multiplicity affects how the polynomial graph behaves at these roots, as it dictates whether the graph crosses or merely touches the x-axis.

In our example function, we see that \(x=3\) is a root with multiplicity 1, meaning it appears only once, while \(x=-6\) has multiplicity 3, appearing three times.

Graph Behavior of Polynomials

The behavior of polynomial graphs is influenced by the degree of the polynomial and the multiplicity of its roots.
These factors affect the way the graph interacts with the x-axis.

• A higher degree polynomial means more turns and complexity, while lower degree polynomials are simpler with fewer turns.
• For instance, a cubical polynomial function will have at least two bends and may resemble an S-shape.

In the example with the polynomial \(f(x)=4(x-3)(x+6)^3\), the overall behavior is shaped by its degree, which is 4 when expanded, due to the \(x-3\) and \(x+6\) terms.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, which involve operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
They take on the general form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants, and \(n\) is a non-negative integer.

• The degree of the polynomial is determined by the highest power of \(x\) and gives insight into the number of roots and the shape of the graph.
• Polynomials are continuous and smooth, meaning no gaps or sharp corners, making their graphs predictable at the basics.

Understanding the structure of polynomials helps in analyzing their behavior, particularly when determining their roots and their graph's interaction with the x-axis.

Crossing and Touching the x-axis

When examining polynomial functions, a critical aspect to understand is how the graph interacts with the x-axis.
This interaction is guided by the multiplicity of the roots.

• If a root has an odd multiplicity, the graph crosses the x-axis at this point.
• If the root has an even multiplicity, the graph merely touches the x-axis and reverses direction.

For example, in the function \(f(x)=4(x-3)(x+6)^3\), the point \(x=-6\) corresponds to a root with multiplicity 3 (odd), so the graph crosses at this point.
Conversely, at \(x=3\), with a multiplicity of 1 (odd as well, but since the power signifies single touch, it behaves differently), the crossing is less abrupt than a higher odd multiplicity which indicates a sharper change.
Understanding these nuances makes it easier to predict the behavior of polynomial function graphs around their zeros.