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Chapter 3: Problem 93

In Exercises \(91-96,\) a polynomial function \(f(x)\) is given in both expanded and factored forms. Graph the function, and solve the equations and inequalities. Give multiplicities of solutions when applicable. \(f(x)=2 x^{4}-9 x^{3}-5 x^{2}+57 x-45\) \(=(x-3)^{2}(2 x+5)(x-1)\) (a) \(f(x)=0\) (b) \(f(x)<0\) (c) \(f(x)>0\)

Short Answer

Expert verified
Roots are \(x = 3\) (Multiplicity 2), \(x = -2.5\), and \(x = 1\). \(f(x) < 0\) for \((-鈭�, -\frac{5}{2})\) and \((1, 3)\).\(f(x)>0\) for \((-\frac{5}{2}, 1)\) and \((3, 鈭�)\).

Step by step solution

01

Understanding the Polynomial

First, note that the polynomial is given in both expanded and factored forms. The factored form is \(f(x) = (x-3)^{2}(2x + 5)(x-1)\). This will help in finding the roots and analyzing the graph.
02

Finding the Roots

Set \(f(x) = 0\) and solve for \(x\). The factored form is easier to solve: \[(x-3)^{2}(2x + 5)(x-1) = 0\]. So, \[x = 3 \text{ (multiplicity 2)}, x = -\frac{5}{2}, \text{ and } x = 1\].
03

Multiplicity of Solutions

Roots and their multiplicities are:\(x = 3\) (multiplicity 2), meaning the graph touches the x-axis and turns around at this point.\(x = -\frac{5}{2}\) (multiplicity 1), meaning the graph crosses the x-axis.\(x = 1\) (multiplicity 1), meaning the graph crosses the x-axis.
04

Graph the Function

Plot the roots on the x-axis. The behavior at each root depends on its multiplicity. At \(x = 3\), since multiplicity is 2, the graph touches but does not cross the x-axis. At \(x = -\frac{5}{2}\) and \(x = 1\), the graph crosses the x-axis.
05

Analyzing Inequalities f(x) = 0

From Step 2, the solutions for \(f(x) = 0\) are \[x = 3, x = -\frac{5}{2}, \text{ and } x = 1\].
06

Analyzing Inequality f(x) < 0

Test intervals divided by the roots to find where the function is negative. The intervals are \((-鈭�, -\frac{5}{2})\), \((-\frac{5}{2}, 1)\), \((1, 3)\), and \((3, 鈭�)\). By testing points within each interval, it can be determined that \(f(x) < 0\) on the intervals \((-鈭�, -\frac{5}{2})\), and \( (1, 3)\).
07

Analyzing Inequality f(x) > 0

Test the same intervals to find where the function is positive. The function is positive on the intervals \((-\frac{5}{2}, 1)\), and \((3, 鈭�)\).

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Key Concepts

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

Factored Form
Polynomials can be given in expanded form or factored form. The factored form is especially useful for solving equations. In the given exercise, the polynomial is presented as:
\( f(x) = (x-3)^{2}(2x + 5)(x-1) \).

This format shows the roots of the equation directly, making it easier to understand and solve. When building the graph of a polynomial, the factored form allows you to pinpoint the x-intercepts quickly.
Roots
The roots of a polynomial are the values of x for which the polynomial equals zero. To find these roots, set the polynomial equal to zero and solve:
\[ (x-3)^{2}(2x + 5)(x-1) = 0 \].

Solving each factor separately gives us:
  • \(x = 3 \) (with multiplicity 2)
  • \( x = -\frac{5}{2} \)
  • \( x = 1 \)
These roots are essential for graphing the polynomial and analyzing its behavior.
Multiplicity
The multiplicity of a root tells us how the graph behaves at that root.
  • Roots with odd multiplicity \((1, 3, 5...)\) mean the graph crosses the x-axis at that point.
  • Roots with even multiplicity \((2, 4, 6...)\) mean the graph touches the x-axis but does not cross it.
In this exercise, we found:
  • \( x = 3 \) with multiplicity 2, so the graph touches the x-axis and turns around at this point.
  • \( x = -\frac{5}{2} \) with multiplicity 1, so the graph crosses the x-axis here.
  • \( x = 1 \) with multiplicity 1, so the graph crosses the x-axis here too.
Graphing Polynomial
Graphing a polynomial helps us visualize its behavior. Here's how to graph it:

  • Plot the roots on the x-axis.
  • Analyze the multiplicity of each root to determine how the graph behaves at each root.
  • Draw the graph, ensuring it touches or crosses the x-axis at the roots based on their multiplicities.
By following these steps, you create an accurate graph of the polynomial.
Inequalities
Inequalities involving polynomials help us understand where the function is positive or negative.
To solve:
  1. Find the roots by solving \( f(x) = 0 \).
  2. Divide the number line into intervals using the roots.
  3. Test points within each interval to determine if the polynomial is positive or negative in that interval.
For example, with the polynomial given:
  • It is negative on \((-\frac{5}{2}, 1)\) and \((1, 3)\).
  • It is positive on \((-\frac{5}{2}, 1)\) and \((3, 鈭�)\).
Understanding these intervals is crucial for solving polynomial inequalities.

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Most popular questions from this chapter

The federal government has developed the body mass index (BMI) to determine ideal weights. A person's BMI is directly proportional to his or her weight in pounds and inversely proportional to the square of his or her height in inches. (A BMI of 19 to 25 corresponds to a healthy weight.) A 6 -foot-tall person weighing 177 Ib has BMI \(24 .\) Find the BMI (to the nearest whole number) of a person whose weight is 130 lb and whose height is 66 in.

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