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Chapter 3: Problem 32

Graph each polynomial function. Factor first if the expression is not in factored form. $$f(x)=x(x+1)(x-1)$$

Short Answer

Expert verified
Roots: \( x = -1, 0, 1 \). Graph crosses the x-axis at all three roots and follows the end behavior of a cubic polynomial.

Step by step solution

01

Understanding the polynomial function

The given function is: \( f(x) = x(x+1)(x-1) \). It's already in factored form. Each factor corresponds to a root of the polynomial.
02

Identify the roots

Set each factor equal to zero to find the roots. \( x = 0, x+1 = 0, x-1 = 0 \) which gives us \( x = 0, x = -1, x = 1 \).
03

Determine the multiplicity of each root

Each root appears only once, so each root has a multiplicity of 1. This means the graph will cross the x-axis at these roots.
04

Plot the roots on the graph

Mark the points \( (-1, 0), (0, 0), (1, 0) \) on the x-axis.
05

Determine end behavior

The leading term of the polynomial is \( x^3 \) (which dominates the behavior at large \( |x| \)). Since the leading coefficient is positive and the degree is odd, the end behavior is: as \( x \to -\infty\), \( f(x) \to -\infty \), and as \( x \to +\infty \), \( f(x) \to +\infty \).
06

Sketch the graph

Draw the graph. Begin from the negative side, passing through the root at \( x = -1 \) (crossing the x-axis), then passing through the origin (crossing the x-axis), and through the root at \( x = 1 \) (crossing the x-axis) and moving upward.

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Key Concepts

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

Factored Form
Factored form is a way of writing polynomial equations so that you can easily identify the roots of the polynomial. It's represented as a product of its factors. For example, the polynomial function given is:
\[ f(x) = x(x+1)(x-1) \] Yes, it's already in factored form! This means it is expressed as a multiplication of simpler polynomials. Each factor can be set to zero to find the roots.
Factored form is incredibly useful for several reasons:
  • It simplifies the process of finding roots.
  • Helps in understanding the structure of the polynomial.
  • Aids in sketching the graph.
    Let's see how it helps in identifying polynomial roots in the next section.
Polynomial Roots
Roots of a polynomial are the x-values where the polynomial equals zero. In this exercise, our polynomial is
\[ f(x) = x(x+1)(x-1) \] To find the roots, set each factor to zero:
  • \( x = 0 \)
  • \( x+1 = 0 \rightarrow x = -1 \)
  • \( x-1 = 0 \rightarrow x = 1 \)
    So, the roots are:
    • x = 0
    • x = -1
    • x = 1
      Now, you plot these roots on the x-axis at points (-1, 0), (0, 0), and (1, 0). The number of times a particular root occurs (multiplicity) affects how the graph behaves at that root. In this case, each root appears only once, so the graph crosses the x-axis at these points.
End Behavior
The end behavior of a polynomial graph describes how the function behaves as \( x \to +\infty \) and\( x \to -\infty \). This can be determined by the leading term of the polynomial. In our example, the leading term comes from multiplying the highest-degree terms of each factor:
\[ x^3 \] Since the leading coefficient (1 in this case) is positive and the degree (3) is odd, our end behavior is:
  • As \( x \to -\infty\), \( f(x) \to -\infty \).
  • As \( x \to +\infty\), \( f(x) \to +\infty \).
    Graphically, this means the polynomial starts from very low on the left (negative x-values) and ends very high on the right (positive x-values), crossing the x-axis at the roots. This understanding helps sketch a more accurate graph of the polynomial function.

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Most popular questions from this chapter

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