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Chapter 0: Problem 55

In Exercises \(55-58\), find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. $$ f(x)=x^{3}-22 x^{2}+17 x+19 $$

Short Answer

Expert verified
Plot the function with initial x-range of -10 to 30, then refine the window settings to make all zeros visible, e.g., x-range of -5 to 25.

Step by step solution

01

- Understand the Function

The function given is a cubic polynomial: ewline f(x) = x^3 - 22x^2 + 17x + 19. ewline The zeros of the function are the values of x where f(x) = 0.
02

- Identify the Degree and Leading Coefficient

The degree of the polynomial is 3, which is the highest power of x, and the leading coefficient is 1 (the coefficient of x^3). This implies that the graph of the function will have one or two turning points.
03

- Determine the Range of x-values

To ensure that all zeros of the polynomial are displayed, consider estimating a reasonable range for the x-values. A good starting point can be to use the fundamental theorem of algebra which suggests there will be 3 roots, real or complex.
04

- Use Graphing Tools

Use a graphing calculator or software to plot the function. Start with a wide window setting, for example, set the x-range from -10 to 30 (based on possible expected zeros) and y-range appropriate to the values of the function within this range.
05

- Refine the Window Setting

After plotting the graph with initial settings, adjust the window settings to zoom in or out until all the zeros are visible. For example, an x-range from -5 to 25 and a suitable y-range may show a clear view of the polynomial's behavior.
06

- Check for All Zeros

Ensure that the graph shows all the points where the function crosses the x-axis. These crossings represent the zeros of the polynomial.

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Key Concepts

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

cubic polynomial
A cubic polynomial is a polynomial of degree three, which means the highest exponent of the variable is three. The general form of a cubic polynomial is \[f(x) = ax^3 + bx^2 + cx + d\]where
  • \(a\), \(b\), \(c\), and \(d\) are constants
  • \(a 鈮� 0\) (since it's cubic)
Cubic polynomials can have up to three real roots and can exhibit one or two turning points, making their graphs more dynamic compared to lower-degree polynomials. Understanding the behavior of a cubic polynomial is the first step to graphing it correctly, as it can have varied shapes. Identifying the turning points and the general trend (whether it opens upwards or downwards) is fundamental for accurate graphing.
finding zeros
Finding the zeros of a polynomial involves solving the equation \(f(x) = 0\). For a cubic polynomial like \(f(x) = x^3 - 22x^2 + 17x + 19\), this means finding the values of \(x\) where the function equals zero. There are several methods to find these zeros:
  • Factoring: Sometimes, a polynomial can be factored into simpler components, although this method is often challenging for cubic polynomials.
  • Graphing: Plotting the function on a graph to visually identify where it crosses the \(x\)-axis.
  • Using a graphing calculator: Graphing tools can provide precise zeroes by using numerical methods.
In our exercise, because factoring is not straightforward, using a graphing calculator is effective. By setting an appropriate window on the graphing calculator, you can identify where the polynomial crosses the \(x\)-axis. Adjusting this window helps to ensure that all roots are visible.
graphing calculator
A graphing calculator is a useful tool for visualizing polynomials and finding their characteristics such as zeros, intercepts, and turning points.To graph a cubic polynomial like \(f(x) = x^3 - 22x^2 + 17x + 19\), follow these steps:
  1. Input the function: Enter the polynomial into the calculator.
  2. Set the window: Choose a suitable \(x\)-range and \(y\)-range. Start with a broad range, e.g., \(-10\ to \30\) for \(x\) and an appropriate range for \(y\).
  3. Plot the graph: Assess the preliminary graph to see the general shape and where it crosses the \(x\)-axis.
  4. Refine the window: Adjust the window settings to zoom in on areas where the function crosses the \(x\)-axis to precisely find all zeros.
  5. Check the zeros: Use the calculator's function to find zeros accurately.
Using a graphing calculator accurately allows students to determine not just the zeros but also understanding the polynomial's overall behavior. This is crucial for more advanced mathematical studies and applications.

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