Problem 55
Question
In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$ f(x)=2 x^{3}+x^{2}-14 x-7 $$
Step-by-Step Solution
Verified Answer
The upper and lower bounds of the zeros for the function can be found by graphing the function and observing the x-intercepts (zeros). The observations can then be verified using the synthetic division method.
1Step 1: Identify the Zeros using Graphing Utility
Firstly, graph the function \(f(x) = 2x^{3} + x^{2} - 14x - 7\). Use a graphing calculator or another graphing tool, and locate the x-intercepts on the graph. These are the zeros of the function.
2Step 2: Determine the Upper and Lower Bounds
Observe the graph to identify where the x-intercepts lie. The smallest x-intercept is the lower bound and the largest x-intercept is the upper bound for the zeros. Assume these as \(lb\) and \(ub\) respectively.
3Step 3: Verify Observations using Synthetic Division
Use synthetic division to verify observations. For the lower bound (lb), all the numbers on the bottom row should be positive or zero. For the upper bound (ub), the numbers on the bottom row should alternate between positive and negative or zero, starting with positive. If not, adjust the bounds accordingly.
Key Concepts
Zeros of PolynomialsGraphing UtilitiesSynthetic DivisionUpper and Lower Bounds
Zeros of Polynomials
The zeros of a polynomial are the values of \( x \) for which the polynomial equals zero. In simpler terms, these are the points where the graph of the polynomial crosses or touches the x-axis. Finding the zeros of a polynomial is an essential part of understanding the overall behavior of the function.
In the context of the given polynomial function \( f(x) = 2x^{3} + x^{2} - 14x - 7 \), the zeros can be considered as solutions to the equation \( f(x) = 0 \). To find these zeros, you might use various methods, such as factoring, using the quadratic formula (for second-degree polynomials), or graphing utilities, especially for higher-degree polynomials.
When examining the function graphically, look for the intersection points with the x-axis. These points will provide an estimate that can then be verified using analytical methods.
In the context of the given polynomial function \( f(x) = 2x^{3} + x^{2} - 14x - 7 \), the zeros can be considered as solutions to the equation \( f(x) = 0 \). To find these zeros, you might use various methods, such as factoring, using the quadratic formula (for second-degree polynomials), or graphing utilities, especially for higher-degree polynomials.
When examining the function graphically, look for the intersection points with the x-axis. These points will provide an estimate that can then be verified using analytical methods.
Graphing Utilities
Graphing utilities are powerful tools for visualizing the behavior of polynomial functions. These tools help in quickly identifying key characteristics such as zeros, extrema, and inflection points. Graphing calculators and computer software like Desmos, GeoGebra, or TI graphing calculators are commonly used to graph these functions.
For the polynomial \( f(x) = 2x^{3} + x^{2} - 14x - 7 \), using a graphing utility involves plotting the function over a certain range. The graph will clearly show the x-intercepts, providing visual evidence of the zeros or roots of the polynomial. Not only do these tools aid in finding zeros, but they also provide a way to explore the function's behavior at different intervals.
When using graphing utilities, it’s crucial to set an appropriate scale to ensure all features of the graph are visible. A graph can also highlight the points of maximum and minimum values, offering a richer understanding of the polynomial's overall shape.
For the polynomial \( f(x) = 2x^{3} + x^{2} - 14x - 7 \), using a graphing utility involves plotting the function over a certain range. The graph will clearly show the x-intercepts, providing visual evidence of the zeros or roots of the polynomial. Not only do these tools aid in finding zeros, but they also provide a way to explore the function's behavior at different intervals.
When using graphing utilities, it’s crucial to set an appropriate scale to ensure all features of the graph are visible. A graph can also highlight the points of maximum and minimum values, offering a richer understanding of the polynomial's overall shape.
Synthetic Division
Synthetic division is a simplified form of polynomial division, particularly useful when dividing by linear factors of the form \( x - c \). It is often employed to verify the zeros obtained from other methods like graphing.
To use synthetic division, arrange the coefficients of the polynomial in descending order of power. Suppose \( f(x) = 2x^{3} + x^{2} - 14x - 7 \) has a suspected zero at \( x = c \). The process will provide a quick check: the remainder should be zero for \( x = c \) to be an actual zero.
This technique is efficient as it requires less written work than traditional polynomial division. The result also tells you about the factorization of the polynomial. When used in conjunction with other methods, synthetic division confirms whether the identified zeros are accurate and reliable.
To use synthetic division, arrange the coefficients of the polynomial in descending order of power. Suppose \( f(x) = 2x^{3} + x^{2} - 14x - 7 \) has a suspected zero at \( x = c \). The process will provide a quick check: the remainder should be zero for \( x = c \) to be an actual zero.
This technique is efficient as it requires less written work than traditional polynomial division. The result also tells you about the factorization of the polynomial. When used in conjunction with other methods, synthetic division confirms whether the identified zeros are accurate and reliable.
Upper and Lower Bounds
The concept of upper and lower bounds is valuable when approximating the zeros of a polynomial. These bounds are estimates to pin down the region on the x-axis where the polynomial's zeros might lie.
When using graphing utilities, you can visually inspect where the x-intercepts occur. The smallest observed x-intercept becomes the lower bound, and the largest becomes the upper bound for the zeros of the polynomial.
For analytical verification, synthetic division helps determine whether these bounds are accurate. The signs of the results from synthetic division indicate if a number functions as an upper or lower bound. For the lower bound, all numbers in the bottom row should be non-negative. To serve as an upper bound, the signs should alternate starting with a positive number. Adjusting these bounds ensures a more accurate localization of the polynomial’s zeros.
When using graphing utilities, you can visually inspect where the x-intercepts occur. The smallest observed x-intercept becomes the lower bound, and the largest becomes the upper bound for the zeros of the polynomial.
For analytical verification, synthetic division helps determine whether these bounds are accurate. The signs of the results from synthetic division indicate if a number functions as an upper or lower bound. For the lower bound, all numbers in the bottom row should be non-negative. To serve as an upper bound, the signs should alternate starting with a positive number. Adjusting these bounds ensures a more accurate localization of the polynomial’s zeros.
Other exercises in this chapter
Problem 54
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