Problem 55 - 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 \(P(x)=x^{3}-2 x^{2}-11 x+12,\). What are the linear factors of \(P(x) ?\)

Step-by-Step Solution

Answer

The linear factors of \( P(x) \) are \( (x - 1), (x - 3), (x + 4) \).

Step 1: Find the Real Zeros

To find the linear factors of the polynomial \( P(x) = x^3 - 2x^2 - 11x + 12 \), we first need to find the real zeros (roots) of the function. These are the values of \( x \) for which \( P(x) = 0 \). We can use the Rational Root Theorem to test possible rational roots, which are the factors of the constant term (12) divided by the factors of the leading coefficient (1). This means the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 2: Evaluate Potential Roots

Evaluate the polynomial \( P(x) \) at the possible rational roots to find which ones make \( P(x) = 0 \). By testing these values: \( P(1) = 0 \), \( P(3) = 0 \), and \( P(-4) = 0 \), we find that the roots are \( x = 1, x = 3, \) and \( x = -4 \).

Step 3: Write the Linear Factors

Each root corresponds to a linear factor of the polynomial. If \( x = 1 \) is a root, then \( (x - 1) \) is a factor; if \( x = 3 \) is a root, then \( (x - 3) \) is a factor; and if \( x = -4 \) is a root, then \( (x + 4) \) is a factor. Thus, the linear factors of \( P(x) \) are \( (x - 1), (x - 3), \) and \( (x + 4) \).

Key Concepts

Real Zeros

Understanding real zeros of a polynomial function is crucial. Real zeros of a polynomial are the values of \( x \) for which the polynomial is equal to zero. In simple terms, these are the points where the graph of the polynomial crosses the x-axis. For example, for the function \( P(x) = x^3 - 2x^2 - 11x + 12 \), the real zeros are the values of \( x \) where \( P(x) = 0 \). These can be found using various methods, such as factoring, graphing, or applying the Rational Root Theorem. Each real zero corresponds to a solution of the equation \( P(x) = 0 \). Discovering these zeros is the first step in understanding the behavior and shape of the graph.

x-intercepts

An x-intercept on a graph is a point where the graph crosses or touches the x-axis. In terms of polynomials, the x-intercepts are exactly the real zeros of the function. These points are significant because they represent the solutions to the equation formed by setting the polynomial equal to zero. For the polynomial \( P(x) = x^3 - 2x^2 - 11x + 12 \), the x-intercepts are the points \( (1, 0), (3, 0), \) and \( (-4, 0) \).

• The x-coordinate of each intercept is a solution to \( P(x) = 0 \).
• x-intercepts help us understand where the graph changes direction across the x-axis.

Finding these intercepts helps visualize the roots in a graphical manner, aiding in comprehensive understanding of the polynomial's behavior.

Rational Root Theorem

The Rational Root Theorem is a valuable tool for finding real zeros of a polynomial when they are rational numbers. It provides a way to identify potential rational roots of a polynomial by considering the factors of the constant term and the leading coefficient. For the polynomial \( P(x) = x^3 - 2x^2 - 11x + 12 \):

• The constant term is 12, and its factors are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).
• The leading coefficient is 1, with the factor \( \pm 1 \).

Therefore, the possible rational roots are all the combinations of these factors over 1. Evaluating the polynomial at these potential roots allows identification of the actual zeros. Using the Rational Root Theorem is efficient, especially with higher-degree polynomials, providing a structured approach to testing potential roots without unnecessary guesswork.