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Chapter 3: Problem 26

Find the \(x\) -intercepts and discuss the behavior of the graph of each polynomial function at its \(x\) -intercepts. $$f(x)=x^{3}-x^{2}-x+1$$

Short Answer

Expert verified
The x-intercepts are x = 1 and x = -1. The graph touches the x-axis at x = 1 and crosses it at x = -1.

Step by step solution

01

- Factoring the Polynomial

First, factor the polynomial function. Consider the given function: \[ f(x) = x^3 - x^2 - x + 1 \] Group the terms and factor by grouping: \[ f(x) = (x^3 - x^2) - (x - 1) \] Factor out the common factors: \[ x^2(x - 1) - 1(x - 1) \] Notice that \(x - 1\) is a common factor: \[ f(x) = (x - 1)(x^2 - 1) \] Next, factor \(x^2 - 1\) as a difference of squares: \[ f(x) = (x - 1)(x - 1)(x + 1) \] Therefore, the original polynomial factors as: \[ f(x) = (x - 1)^2 (x + 1) \]
02

- Finding the x-intercepts

To find the x-intercepts, set the factored polynomial equal to zero: \[ (x - 1)^2 (x + 1) = 0 \] Solve for \(x\): \[ (x - 1)^2 = 0 \] \[ x = 1 \] and \[ (x + 1) = 0 \] \[ x = -1 \] So, the x-intercepts are \(x = 1\) and \(x = -1\).
03

- Analyzing the Behavior at the x-intercepts

Now, analyze the behavior of the graph at each x-intercept: For \(x = 1\): Since \((x - 1)\) is squared to \( (x - 1)^2 \), this indicates that the graph touches the x-axis at \(x = 1\) but does not cross it. This is because multiplicity is even. For \(x = -1\): The factor \((x + 1)\) indicates that the graph crosses the x-axis at \(x = -1\). This is because multiplicity is odd.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercepts
An x-intercept is where a graph crosses the x-axis. To find the x-intercepts of a polynomial, set the polynomial equal to zero and solve for x. In the given function \( f(x) = x^3 - x^2 - x + 1 \), we factored it to \(f(x) = (x - 1)^2 (x + 1)\). Setting this to zero, we get the solutions: \((x - 1)^2 = 0\) and \(x + 1 = 0\). These give us the x-intercepts at \( x = 1 \) and \( x = -1 \). Each intercept corresponds to a point where the graph meets the x-axis.
factoring polynomials
Factoring polynomials involves expressing them as a product of simpler polynomials. For \( f(x) = x^3 - x^2 - x + 1 \), we factor by grouping, yielding \(f(x) = (x - 1)(x - 1)(x + 1)\). This simplifies to \(f(x) = (x - 1)^2 (x + 1)\). Factoring helps break down complex polynomials into manageable parts and is crucial for finding x-intercepts. Notice how recognizing common factors and patterns like the difference of squares makes this easier.
behavior at intercepts
The behavior of the graph at x-intercepts depends on the multiplicity of each intercepted factor. For \(f(x) = (x - 1)^2 (x + 1)\), when \( x = 1 \), the \((x - 1)^2\) term indicates the graph only touches the x-axis without crossing it because the multiplicity is even. For \(x = -1\), the \( (x + 1) \) term has a multiplicity of 1, which is odd, indicating that the graph crosses the x-axis at this point. The multiplicity tells us whether the graph will simply touch or fully cross the axis at each intercept.
multiplicity
Multiplicity refers to the number of times a particular factor appears in the factored form of the polynomial. In \(f(x) = (x - 1)^2 (x + 1)\), the term \(x - 1\) has a multiplicity of 2. At \(x = 1\), the graph touches the x-axis but doesn't cross it because the multiplicity is even. In contrast, the term \(x + 1\) has a multiplicity of 1. At \(x = -1\), the graph crosses the x-axis because the multiplicity is odd. Understanding multiplicity helps predict the graph's behavior at each intercept.

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