Problem 64 - Algebra for College Students

Question

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only two \(x\) -intercepts at \((5,0)\) and \((-6,0)\) It does not cross the \(x\) -axis at either \(x\) -intercept.

Step-by-Step Solution

Answer

f(x) = (x - 5)^2(x + 6)^2

Step 1 - Understand the problem

The graph of the polynomial must touch the x-axis at the points \(5, 0\) and \(-6, 0\), but not cross it. This implies that these points are double roots of the polynomial.

Step 2 - Define the roots

Since the polynomial touches but does not cross the x-axis at \(5, 0\) and \(-6, 0\), the roots must be of even multiplicity. Let’s assume they are of multiplicity 2.

Step 3 - Form the polynomial

Given roots \(5, 0\) and \(-6, 0\) with multiplicity 2, the polynomial can be written as: \( f(x) = a(x - 5)^2(x + 6)^2 \) where \(a\) is a non-zero constant.

Step 4 - Choose a value for a

Typically \(a = 1 \) is chosen for simplicity unless specified otherwise. Therefore, \( f(x) = (x - 5)^2(x + 6)^2 \).

Key Concepts

Double Roots

In polynomial functions, double roots are quite interesting. They occur when the graph of the polynomial touches the x-axis at a particular point but doesn't cross it.
This happens because the root is repeated, meaning the factor corresponding to that root is squared.
For example, consider the polynomial from the exercise:
\[ f(x) = (x - 5)^2(x + 6)^2 \]
Here, 5 and -6 are double roots because each one is squared in the polynomial.
When the graph hits these points, it simply touches the x-axis and bounces back instead of crossing it.

X-Intercepts

The x-intercepts of a polynomial are the points where the graph intersects the x-axis.
These points can be found by setting the polynomial equal to zero and solving for x.
In our specific exercise, the x-intercepts are given as (5, 0) and (-6, 0).
This means the polynomial is zero when x is 5 or -6.
\[ f(x) = (x - 5)^2(x + 6)^2 \]
At these points, the graph touches the x-axis but doesn't cross, indicating double roots with an even multiplicity.

Multiplicity

Multiplicity refers to the number of times a particular root is repeated in a polynomial.
A root with multiplicity 2 means that the factor is squared in the polynomial.
For example, in our exercise, both x = 5 and x = -6 have a multiplicity of 2.
\[ f(x) = (x - 5)^2(x + 6)^2 \]
Higher multiplicities also affect how the graph behaves around that root:

• A root with an odd multiplicity means the graph crosses the x-axis at that point.
• A root with an even multiplicity means the graph just touches the x-axis but doesn't cross it.

Understanding multiplicity is key to predicting the shape of the polynomial's graph around its roots.