Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Answer true or false to each statement. Then support your answer by graphing. If a polynomial function of even degree has a negative leading coefficient and a positive \(y\) -value for its \(y\) -intercept, it must have at least two real zeros.

Step-by-Step Solution

Answer

True. At least two real zeros are needed due to the graph's behavior.

Step 1: Understand the Characteristics of the Polynomial

A polynomial function of even degree with a negative leading coefficient has the end behavior where as \(x\) approaches positive or negative infinity, \(f(x)\) goes to negative infinity (both ends down). This characteristic is crucial to determine the number of real zeros.

Step 2: Assess the y-intercept

The statement indicates that the \(y\)-intercept is positive. This means that the graph of the polynomial must cross the \(y\)-axis above the x-axis.

Step 3: Analyze Possible Zeros from End Behavior and y-intercept

Given the end behavior that both ends point downwards and it crosses the \(y\)-axis above it, the graph will have to turn downwards again to satisfy its overall end behavior. This suggests the potential existence of at least one local maximum.

Step 4: Use Graph to Confirm Zeros

To have a local maximum and then go back down, the graph must intersect the x-axis at least twice. Therefore, it must have at least two real zeros, which corresponds to two points where it crosses the \(x\)-axis.

Key Concepts

Even Degree Polynomials

Polynomial functions can have various degrees, which indicate the highest power of the variable present in the polynomial. Even degree polynomials, such as quadratic (\(x^2\)), quartic (\(x^4\)), and more generally \(x^{2n}\) for natural numbers \(n\), have special characteristics.
They have ends that move in the same direction on the graph. This means that as \(x\) approaches positive or negative infinity, the values of \(f(x)\) either both rise infinitely or both fall infinitely.
This behavior is consistent unless influenced by the polynomial's leading coefficient. These consistent end behaviors in even degree polynomials make them unique in graphing and analyzing solutions.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It significantly influences the graph of a polynomial.
For polynomials of even degree:

• If the leading coefficient is positive, the graph tends to both ends up (as \(x\) approaches positive or negative infinity, \(f(x)\) goes to positive infinity).
• If the leading coefficient is negative, the graph tends to both ends down (as \(x\) approaches positive or negative infinity, \(f(x)\) goes to negative infinity).

This characteristic impacts the number of potential real zeros and graphical intercept with the x-axis.
Understanding how the leading coefficient plays into polynomial curve behavior aids in predicting and visualizing polynomial solutions.

Real Zeros

Real zeros of a polynomial function are the x-values where the polynomial equals zero, meaning these are the points where the graph intersects the x-axis.
They are critical in defining the behavior of polynomial functions.

• Real zeros are also referred to as roots or x-intercepts and can be verified visually through graphing.
• The number of real zeros in a polynomial can be influenced by the degree of the polynomial and the coefficients.

In even degree polynomials with a negative leading coefficient and a positive y-intercept, the real zeros must logically exist in pairs.
This is because one x-intercept requires a return back to the x-axis, creating another real zero to satisfy the polynomial's end behavior.

Graphing Polynomials

Graphing polynomials is a crucial method for understanding their behavior, especially in determining zeros and intercepts.
When graphing:

• Start by plotting the y-intercept, the point where \(x = 0\).
• Use the leading coefficient to determine end behavior—whether both ends point upward or downward.
• Determine and plot the x-intercepts or zeros to understand where the graph crosses or touches the x-axis.

This process helps visualize the potential number of real zeros. For instance, even degree polynomial functions with both ends down and a positive y-intercept must turn back down, confirming the presence of at least two real zeros.
Graphing aids significantly in conceptualizing the theoretical behavior of polynomial functions.