Problem 59 - A Graphical Approach to College Algebra

Question

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). Give the solution set of \(P(x)>0,\) using interval notation.

Step-by-Step Solution

Answer

The solution set is \((-3, 1) \cup (4, \infty)\).

Step 1: Identify Real Zeros

To find the intervals where the function is greater than zero, we first need to find the real zeros of the polynomial. These zeros are critical because they are the points where the function changes its sign. We solve the equation \(P(x) = x^3 - 2x^2 - 11x + 12 = 0\).

Step 2: Find Real Zeros

To solve for the real zeros, we can use factoring or numerical methods like the Rational Root Theorem. Testing possible rational roots, we find that \(x = 1, -3, 4\) are zeros of the polynomial. So, the function \(P(x)\) can be factored as \((x - 1)(x + 3)(x - 4) = 0\).

Step 3: Determine Sign Changes

Now, we divide the number line into intervals based on the zeros: \((-\infty, -3)\), \((-3, 1)\), \((1, 4)\), and \((4, \infty)\). We will test the sign of \(P(x)\) in each of these intervals to determine where it is positive.

Step 4: Test Intervals

Choose test points in each interval:- Interval \((-\infty, -3)\), e.g., use \(x = -4\). Plugging into \(P(x)\) gives a negative value.- Interval \((-3, 1)\), e.g., use \(x = 0\). Plugging into \(P(x)\) gives a positive value.- Interval \((1, 4)\), e.g., use \(x = 2\). Plugging into \(P(x)\) gives a negative value.- Interval \((4, \infty)\), e.g., use \(x = 5\). Plugging into \(P(x)\) gives a positive value.

Step 5: Write the Solution

Based on the tests, the function \(P(x)\) is positive in the intervals \((-3, 1)\) and \((4, \infty)\). Therefore, the solution set for \(P(x) > 0\) is the union of these intervals.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the x-values where the function equals zero. These zeros are also known as roots of the equation. They are found by solving the equation \(P(x) = 0\). For the given polynomial \(P(x) = x^3 - 2x^2 - 11x + 12\), the goal is to determine the x-values at which the graph will cross or touch the horizontal axis.

To find real zeros, we can use methods such as factoring, synthetic division, or applying the Rational Root Theorem. In this case, using these methods, we find the roots \(x = 1, -3,\) and \(4\). These real zeros imply that the function changes its sign at these points, forming critical points for analyzing the graph's behavior around them.

Let's reflect on the significance of these zeros:

• They provide insight into where the function might shift from positive to negative values or vice versa.
• The real zeros help in breaking the x-axis into segments, creating intervals for further analysis.

Graphing Functions

Graphing a polynomial function gives a visual representation of its behavior across different values of \(x\). The zeros we find in polynomial equations are the points where the graph intersects the x-axis. For \(P(x) = x^3 - 2x^2 - 11x + 12\), the graph will intersect at \(x = 1\), \(-3\), and \(4\).

When you graph the polynomial, these intersection points serve as guides to sketch the overall shape, including lobes and curves. Here are a few key considerations when graphing polynomial functions:

• Determine and plot the zeros on the graph.
• Identify the end behavior, which depends on the leading term's degree and sign (in this case, \(x^3\) suggests a typical cubic shape).
• Evaluate the behavior around the zeros, checking whether the curve passes through or just touches the x-axis.

Graphing functions not only helps visualize solutions but also aids in identifying positive and negative intervals.

Interval Notation

Interval notation is a mathematical shorthand used to express a range of values. It's especially useful when describing the domain, range, or solutions where a function meets specific criterion, such as being greater than zero. When using interval notation:

• "(" or ")" denotes that the endpoint is not included (open interval).
• "]" or "[" means the endpoint is included (closed interval).

For the example polynomial \(P(x) > 0\), determining where the function is positive involves testing each segment between the zeros:

• The function is negative from \((-fty, -3)\).
• In the interval \((-3, 1)\), it gives a positive value.
• Between \((1, 4)\), the result is negative again.
• Positive values reoccur in \((4, fty)\).

Thus, the solution for \(P(x) > 0\) is expressed as the intervals \((-3, 1)\cup(4, fty)\). Interval notation succinctly presents these non-overlapping ranges where the function holds the desired property.

Polynomial Functions

Polynomial functions are expressions that include variables raised to whole number powers, with coefficients on each term. They are fundamental in algebra and calculus, serving as building blocks for more advanced mathematics.

The general form is \(a_nx^n + a_{n-1}x^{n-1} + \, \cdots \, + a_1x + a_0\), where \(a_n\) to \(a_0\) are constants and \(n\) is a non-negative integer, denoting the degree of the polynomial. For example, \(P(x) = x^3 - 2x^2 - 11x + 12\) is a third-degree polynomial because the highest power of \(x\) is 3.

Some essential characteristics of polynomial functions include:

• The degree indicates the maximum number of zeros or x-intercepts the graph can have.
• Polynomials are continuous and smooth curves, without breaks or sharp corners.
• The leading term's degree and coefficient determine the end behavior—how the graph behaves as \(x\) approaches infinitely large positive or negative values.

Understanding these concepts helps in analyzing their graphs, predicting behavior, and solving equations.