Problem 47 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why there cannot be two different points at which the graph of a cubic polynomial touches the \(x\) -axis without crossing it.

Step-by-Step Solution

Answer

A cubic polynomial's graph cannot touch the x-axis at two different points without crossing because this would imply two roots of even multiplicity. However, a cubic polynomial, due to its total root multiplicity being odd (3), cannot support two even multiplicities.

Step 1: Multiplicity of Roots

For a given polynomial, a root is said to be of even multiplicity if the exponent on its factor in the factored form of the polynomial is an even number. Similarly, a root is of odd multiplicity if the exponent is odd. If a root is of even multiplicity, the graph touches the x-axis without crossing at the point corresponding to that root. For an odd multiplicity, the graph crosses the x-axis.

Step 2: Count the Number of Roots

A cubic polynomial \( p(x) = ax^3 + bx^2 + cx + d \) can have at most three real roots. The roots can have combinations such as three distinct real roots, two distinct roots and one repeated root, or three identical roots.

Step 3: Analyze Odd and Even Multiplicity

For the graph to touch the x-axis without crossing it at two distinct points, we would need two roots with even multiplicity. But given the total exponent is a 3 (cubic polynomial), we cannot have two root multiplicities that are both even since the sum of two even numbers can never be an odd number.

Step 4: Resultant Argument

Thus, it can be inferred that a cubic polynomial's graph cannot touch the x-axis at two different points without crossing it. This is because the total sum of root multiplicities cannot be even in a cubic polynomial.

Key Concepts

Multiplicity of Roots

When it comes to understanding the multiplicity of roots in a polynomial, it is important to note how this affects the graph of the polynomial. For any polynomial, each root is associated with a certain multiplicity, which represents the number of times the root appears. This is depicted by the exponent in the factored form of the polynomial.

The multiplicity of a root determines how the polynomial's graph interacts with the x-axis. Specifically:

• If a root has an **even multiplicity**, the graph will touch the x-axis but not cross it at that point. For example, if a root appears twice, it means that the graph "bounces" off the axis.
• If a root has an **odd multiplicity**, the graph crosses the x-axis. This is what happens when a root appears once, or any odd number of times.

This distinction is crucial because it ensures the graph behaves in specific ways at each root, ultimately affecting the entire shape and intersection of the graph with the x-axis.

Real Roots

In a cubic polynomial of the form \( p(x) = ax^3 + bx^2 + cx + d \), real roots are the values of \( x \) for which \( p(x) = 0 \). A cubic polynomial can have different combinations of real roots:

• **Three distinct real roots**: Each root is unique, and the graph will cross the x-axis at three different points.
• **Two distinct real roots and one repeated root**: This means one of the roots appears more than once (even multiplicity), causing the graph to touch the x-axis at this point only once without crossing.
• **Three identical real roots**: Here, the same root appears three times, leading to one point of contact with the x-axis.

The configuration of real roots is dictated by the multiplicity of each. In a cubic polynomial, the sum of root multiplicities must always equal three as this corresponds to the degree of the polynomial.

Graph of Polynomial

The graph of a cubic polynomial provides a visual insight into the behavior of the polynomial related to its roots and their multiplicities. Visualizing the graph helps to understand the polynomial’s interaction with the x-axis:

• When the polynomial has all **odd multiplicity roots**, the graph will intersect the x-axis at each root point.
• If there is a **root with even multiplicity**, the graph will touch but not cross the x-axis at that particular root.

Cubic polynomials can have a few different appearances, largely influenced by their coefficients and roots. However, one important rule is that the graph of a cubic polynomial cannot touch the x-axis at two points without crossing. This is due to the nature of odd degree polynomials. The sum of the multiplicities will always be odd, hence not allowing for two even multiplicities. Therefore, if a cubic polynomial touches the x-axis, it can only do so at a single point or multiple points with crossing.