Problem 2 - College Algebra with Corequisite Support

Question

If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?

Step-by-Step Solution

Answer

The graph of the function intersects the x-axis at \(n\) distinct points.

Step 1: Understand the Relationship between Zeros and the Polynomial

When a polynomial function of degree \(n\) has \(n\) distinct zeros, it means that the function can be expressed as \(f(x) = a(x - r_1)(x - r_2)\cdots(x - r_n)\), where \(a\) is a non-zero constant and \(r_1, r_2, \ldots, r_n\) are the distinct zeros.

Step 2: Analyze the Graph's Intersection with the X-Axis

Each zero \(r_i\) corresponds to an \(x\)-value where the graph of the polynomial intersects the x-axis. Thus, for a polynomial of degree \(n\) with \(n\) distinct zeros, the graph will intersect the x-axis \(n\) times, precisely at each zero \(r_1, r_2, \ldots, r_n\).

Step 3: Determine the Behavior Around the Zeros

At each zero, the polynomial crosses the x-axis since the zeros are distinct. This implies that the sign of the function changes at each zero, resulting in the graph crossing the x-axis at each distinct zero.

Step 4: Understand the Shape of the Polynomial Graph

The general shape of the graph is an \(n\)-degree polynomial, which means its end behavior is determined by the leading term \(a(x)^n\). If \(a > 0\), as \(x\to \pm\infty\), the graph rises from the left and rises on the right for even \(n\), and falls to the left and rises to the right for odd \(n\); the reverse is true for \(a < 0\).

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental aspect when analyzing polynomial functions. It refers to the highest power of the variable in the polynomial expression. For example, in the polynomial \(2x^3 + 3x^2 - x + 5\), the degree is 3 because the highest power of \(x\) is 3.
This degree tells us several important things about the polynomial:

• Number of Zeros: A polynomial of degree \(n\) has at most \(n\) zeros. Zeros are the values of \(x\) where the polynomial equals zero.
• Shape of the Graph: The degree influences the overall shape and the end behavior of the graph.

Understanding the degree helps us predict how many times the polynomial graph will intersect the x-axis. This is directly tied to the roots, or zeros, of the polynomial.

Zeros of a Polynomial

Zeros of a polynomial, also known as roots, are crucial points where the graph of the polynomial intersects the x-axis. For a polynomial function like \(f(x) = a(x - r_1)(x - r_2)\cdots(x - r_n)\), the zeros are \(r_1, r_2, \ldots, r_n\). Each zero represents an x-value for which \(f(x) = 0\).

• Distinct Zeros: If a polynomial of degree \(n\) has \(n\) distinct zeros, it means the graph intersects the x-axis at \(n\) unique points.
• Multiplicity of Zeros: If a zero has a multiplicity greater than one, the polynomial might touch the x-axis but not cross it completely at that zero.

Finding zeros allows us to factor the polynomial fully and understand where its graph lies with respect to the x-axis.

Graph of a Polynomial

The graph of a polynomial gives a visual representation of the polynomial equation. One can intuitively understand many properties of the polynomial through its graph. For instance, a polynomial with \(n\) distinct zeros will intersect the x-axis precisely \(n\) times.

• Intersection Points: These intersections occur at each zero of the polynomial. At these points, the graph will move from one side of the x-axis to the other if the zeros are distinct.
• Turning Points: A polynomial of degree \(n\) can have up to \(n-1\) turning points, places where the graph changes direction.

The graph allows us to see the behavior around the zeros and understand how the polynomial function behaves at various ranges of x-values.

End Behavior of Polynomials

The end behavior of a polynomial describes how the graph behaves as \(x\) approaches positive or negative infinity. This behavior is majorly influenced by the leading term, \(a(x)^n\), where \(a\) is the leading coefficient, and \(n\) is the degree.

• Even vs. Odd Degree: For even-degree polynomials, if the leading coefficient \(a > 0\), the graph rises on both sides, while if \(a < 0\), it falls. For odd-degree polynomials, if \(a > 0\), the graph falls on the left and rises on the right, and the opposite is true if \(a < 0\).
• Determining the End: By examining the leading term, one can quickly determine the end behavior and predict how the graph stretches out infinitely.

Understanding end behavior is essential for sketching the graph and predicting how the function behaves in extreme values, helping to contextualize the polynomial in real-world scenarios.