Problem 52 - Calculus

Question

In Problems \(52-57\), give an example of: A polynomial of degree 3 whose graph cuts the horizontal axis three times to the right of the origin.

Step-by-Step Solution

Answer

An example is the polynomial \( f(x) = (x - 1)(x - 2)(x - 3) \).

Step 1: Understand the Problem

The problem asks for a polynomial of degree 3 that crosses the x-axis three times to the right of the origin. This means we are looking for a cubic function with three distinct positive roots.

Step 2: Form the General Polynomial

A cubic polynomial can be expressed in the form \( f(x) = a(x - r_1)(x - r_2)(x - r_3) \), where \( r_1, r_2, \) and \( r_3 \) are the roots. Since we want the roots to be all positive, choose \( r_1, r_2, \) and \( r_3 \) such that each is positive.

Step 3: Choose Specific Roots

Select specific positive values for the roots. For example, let \( r_1 = 1 \), \( r_2 = 2 \), and \( r_3 = 3 \).

Step 4: Write the Polynomial

Substitute the chosen roots into the polynomial form. Therefore, the polynomial can be given as \( f(x) = a(x - 1)(x - 2)(x - 3) \).

Step 5: Define the Leading Coefficient

Select a simple value for \( a \) to fully define the polynomial. The simplest choice is \( a = 1 \). This gives \( f(x) = (x - 1)(x - 2)(x - 3) \).

Step 6: Expand the Polynomial (Optional)

Expand \( f(x) = (x - 1)(x - 2)(x - 3) \) to find its standard form. Calculate as follows:\[ f(x) = (x - 1)(x - 2)(x - 3) \]First, multiply the first two factors:\[ = [(x - 1)(x - 2)](x - 3) = (x^2 - 3x + 2)(x - 3) \]Now, expand:\[ = x^3 - 3x^2 + 2x - 3x^2 + 9x - 6 \]Combine like terms:\[ = x^3 - 6x^2 + 11x - 6 \].

Key Concepts

Polynomial Degree

In algebra, a polynomial is made up of terms, each consisting of a coefficient multiplied by the variable raised to a non-negative integer exponent. The degree of a polynomial is the highest exponent of the variable present in the expression.
If you have a polynomial of degree 3, like the one described in this problem, it is called a "cubic polynomial." Cubic polynomials are important because they are the simplest kind of polynomial that can have multiple x-axis intersections or roots.

• A polynomial of degree 3 always has exactly three roots, although they may not be distinct or all real.
• It has the general form: \(f(x) = ax^3 + bx^2 + cx + d\), where \(a eq 0\).

This polynomial can show a variety of behaviors in terms of its shape and intersections with the x-axis, especially when you analyze its roots. Understanding the degree of a polynomial helps in predicting the number of zeroes (roots) it could have, which is crucial for graphing the expression.

Positive Roots

One of the fascinating aspects of cubic polynomials is their roots, or zeroes, which indicate where the polynomial's graph intersects the x-axis. When the problem requires that the polynomial has positive roots, this means all the places where the graph crosses the x-axis are to the right of zero.
Roots can be:

• Real or complex
• Positive or negative

For the solution here, we particularly focus on real, positive roots.
In this exercise, a polynomial is sought to intersect the axis at three positive values. This translates into selecting positive numbers for roots \(r_1, r_2,\) and \(r_3\). For example, if you choose roots such as \(r_1 = 1\), \(r_2 = 2\), and \(r_3 = 3\), the cubic polynomial can be constructed as \(f(x) = (x - 1)(x - 2)(x - 3)\).
Each root shifts the curve right of the origin, making all its intersections with the x-axis positive.

X-axis Intersections

The x-axis intersections of a polynomial graph are simply the points where the graph crosses the x-axis. These intersections correspond to the roots of the polynomial.
A cubic polynomial will have at most three x-axis intersections because it is defined by a degree 3 expression, having three factors or roots.

• The intersection points tell us the values of \(x\) at which \(f(x) = 0\).
• In this context, it means solving the equation \(a(x - r_1)(x - r_2)(x - r_3) = 0\), which gives the values \((r_1, r_2, r_3)\).

For the exercise, what matters is that all these \(r\) values must be greater than zero, ensuring that the polynomial touches the x-axis at positive points.
This visually corresponds to the graph of the polynomial going down to meet the x-axis from the positive side, crossing it, and then moving back up.