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Chapter 3: Problem 42

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{3}+10 x^{2}+33 x+34$$

Short Answer

Expert verified
The zeros of the function are \(x = -1, -2, -17\). In factorized form, the polynomial is \((x + 1)(x + 2)(x + 17)\). These zeros correspond to the x-intercepts of the function's graph, which can be confirmed using a graphing tool.

Step by step solution

01

Finding the Zeros of the Function

In order to find the zeros of the function, set \(f(x) = 0\) and solve for \(x\). This gives us the following equation to solve: \(x^{3}+10 x^{2}+33 x+34 =0\). By using either factorization or synthetic division, we will be able to find the zeros of the function.
02

Factorizing the Polynomial

When we factorize the provided polynomial, we get \((x + 1)(x + 2)(x + 17)=0\). These are the linear factors of the polynomial.
03

Determining the x-intercepts from the Factors

The x-intercepts of the function can be determined from the factors. Setting each factor equal to zero and solving for \(x\) gives \(x = -1, -2, -17\). These are the three x-intercepts of the function.
04

Verifying the x-intercepts using a Graphing Tool

To verify that these are the real zeros and the only x-intercepts, we can use a graphing tool to graph the function \(f(x)\) and observe where it intersects with the x-axis. The graphing tool should show that the function intersects the x-axis at the points we identified.

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Key Concepts

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

Finding Zeros
Finding the zeros of a polynomial function is one of the fundamental tasks in algebra. Zeros are the values of the variable, usually denoted as \(x\), for which the polynomial evaluates to zero. For a given polynomial \(f(x) = x^3 + 10x^2 + 33x + 34\), we find its zeros by setting \(f(x) = 0\) and solving the equation.
  • This involves looking for values of \(x\) that make the expression equal zero.
  • Zeros are critical as they provide points where the graph of the polynomial touches or crosses the x-axis.
  • These points help in understanding the graph's behavior and structure.
Often, to discover the zeros, methods like factorization or synthetic division are employed. In simpler polynomials, these methods help simplify the equation to easily find values of \(x\) that fulfill the zero-condition.
Factorization
Factoring a polynomial means expressing it as a product of its linear components. This process helps reveal the zeros of the function in a more straightforward manner.
For the polynomial \(f(x) = x^3 + 10x^2 + 33x + 34\), factorization shows it can be written as \((x+1)(x+2)(x+17) = 0\).
  • Each component of the factored form, \((x + a)\), corresponds to potential solutions for the function when set to zero.
  • By solving \((x+1)=0\), \((x+2)=0\), and \((x+17)=0\), we find the zeros \(x = -1, -2, \text{ and } -17\).
Factorization not only helps in solving polynomials more efficiently but also simplifies verification of calculations. When a polynomial is expressed in its factored form, identifying solutions is direct and avoids complex algebraic manipulation.
x-intercepts
In mathematics, especially in graphing, x-intercepts are crucial points where the graph of a function intersects the x-axis. These intercepts occur at the zeros of the function. For the polynomial \(f(x)\), the x-intercepts are calculated from the factors obtained during factorization.
For the given example, setting \((x+1)(x+2)(x+17)=0\), the x-intercepts are found to be:
  • \(x = -1\)
  • \(x = -2\)
  • \(x = -17\)
Each x-intercept represents a point where the polynomial equals zero and crosses or touches the x-axis. These points are essential for sketching the function and understanding its general shape and direction.
Graphing Utility
Using a graphing utility or tool is an excellent way to visually verify the solutions obtained from algebraic manipulations. Graphing tools plot the function across various values of \(x\) and allow us to see where it intersects the x-axis.
When we graph \(f(x) = x^3 + 10x^2 + 33x + 34\), we should see the curve crossing the x-axis at the points \(x = -1, -2, \text{ and } -17\).
  • The graph provides a visual confirmation that the calculated zeros are indeed the x-intercepts of the polynomial.
  • It also helps verify that there are no additional points where the graph meets the x-axis.
Graphing utilities enhance understanding by translating numerical and algebraic solutions into a visual format, ensuring a robust verification method. They are fundamental tools both for checking work and for educational purposes.

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Most popular questions from this chapter

A driver averaged 50 miles per hour on the round trip between Baltimore, Maryland, and Philadelphia, Pennsylvania, 100 miles away. The average speeds for going and returning were \(x\) and \(y\) miles per hour, respectively. (a) Show that \(y=\frac{25 x}{x-25}\) (b) Determine the vertical and horizontal asymptotes of the function. (c) Use a graphing utility to complete the table. What do you observe? (d) Use the graphing utility to graph the function. (e) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

The numbers of employees \(E\) (in thousands) in education and health services in the United States from 1960 through 2013 are approximated by \(E=-0.088 t^{3}+10.77 t^{2}+14.6 t+3197\) \(0 \leq t \leq 53,\) where \(t\) is the year, with \(t=0\) corresponding to \(1960. (a) Use a graphing utility to graph the model over the domain. (b) Estimate the number of employees in education and health services in \)1960 .\( Use the Remainder Theorem to estimate the number in \)2010 .$ (c) Is this a good model for making predictions in future years? Explain.

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(3(x-5)<4 x-7\)

A rectangular region of length \(x\) and width \(y\) has an area of 500 square meters. (a) Write the width \(y\) as a function of \(x .\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the function and determine the width of the rectangle when \(x=30\) meters.

Use a graphing utility to graph the function and find its domain and range. $$f(x)=-|x+9|$$
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