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Chapter 10: Problem 63

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only two \(x\) -intercepts at \((-2,0)\) and \((1,0) .\) It crosses the \(x\) -axis at \((-2,0)\) but does not cross at \((1,0)\).

Short Answer

Expert verified
The polynomial function is (x+2)(x-1)^2.

Step by step solution

01

Understand the Problem

We need to find a polynomial function that has exactly two x-intercepts: one at (-2,0) and one at (1,0). The graph crosses the x-axis at (-2,0) but touches (does not cross) the x-axis at (1,0).
02

Determine the Factors

For the graph to intercept the x-axis at (-2,0) and cross it, the factor (x + 2) must appear an odd number of times. For the graph to touch the x-axis at (1,0) without crossing it, the factor (x - 1) must appear an even number of times.
03

Construct the Polynomial

Based on the factors identified, construct the polynomial. Since the factor (x + 2) needs to appear an odd number of times, we can use (x + 2) just once. For (x - 1) to appear an even number of times, we will square it. The polynomial function can be written as (x + 2)(x - 1)^2.
04

Expand the Polynomial (Optional)

Expand the polynomial for a standard form expression. Multiply (x + 2) by (x - 1)^2. (x - 1)^2 = x^2 - 2x + 1. Now multiply: (x + 2)(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2. Simplify: x^3 - 3x + 2.
05

Verify the Solution

Check that the polynomial (x + 2)(x - 1)^2 behaves as required. It has x-intercepts at (-2,0) and (1,0). It crosses the x-axis at (-2) and touches (does not cross) at (1,0) as demanded.

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Key Concepts

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

x-intercepts
The x-intercepts of a polynomial are points where the graph of the polynomial crosses or touches the x-axis. For a point \(a\) to be an x-intercept, the polynomial must have a factor \((x - a)\). This means at \(x = a,\) the value of the polynomial is zero. In our example, the polynomial has x-intercepts at \((-2,0)\)and \(1,0)\). The factors for these intercepts are \((x + 2)\) and \((x - 1)\) respectively. Understanding these intercepts is crucial as they help us know where the polynomial will interact with the x-axis.
graph behavior
The behavior of the graph of a polynomial near its x-intercepts can vary. Sometimes the graph will cross the x-axis, and other times it will just touch it and turn around. How the graph behaves at these points is determined by the multiplicity of the factor associated with the x-intercept. In our exercise, we needed the graph to cross the x-axis at \((-2,0)\) and touch the x-axis at \((1,0)\). If a factor appears an odd number of times in the polynomial, the graph crosses the x-axis at that factor. If it appears an even number of times, the graph touches but does not cross the axis.
polynomial factors
To form a polynomial with specific x-intercepts and defined behavior at those intercepts, we use factors of the polynomial. Each x-intercept \(a\) corresponds to a factor \((x - a)\). If we want the graph to cross the x-axis at \(a\), the factor \((x - a)\) must appear an odd number of times. If we need the graph to touch the x-axis but not cross it at \(a\), the factor must appear an even number of times. In our textbook solution, \((x + 2)\) appears once (crosses at \(-2,0)\), and \((x - 1)^2\) appears twice (touches but does not cross at \((1,0)\).
crossing points
Crossing points occur where the graph of the polynomial intersects and passes through the x-axis. This happens when the associated factor for the x-intercept appears an odd number of times in the polynomial. For our polynomial, \((x+2)\) is a factor that appears once. Therefore, the graph crosses the x-axis at \(-2.\) This crossing tells us that one part of the polynomial graph is above the x-axis and another part is below right at that intercept.
touching points
Touching points occur where the graph of the polynomial just touches the x-axis and does not cross it. This happens when the factor for the x-intercept appears an even number of times. In our example, \((x-1)\^2\) is the relevant factor. It indicates that the graph will touch the x-axis at \(1\) but will not cross it. This point is also known as a 'root' or a 'zero' of the polynomial, where the graph is tangent to the x-axis.

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Most popular questions from this chapter

Let \(f(x)=x^{4}-1, g(x)=x^{3}-3 x^{2}+5,\) and \(h(x)=4 x^{4}-\) \(3 x^{2}+3 x-1 .\) Find the following function values by using two different methods. See Example \(I\) $$ g(-1) $$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+x-2}$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}+4 x+4}$$

Sketch the graph of each pair of functions in the same coordinate system. What do you observe in each case? $$f(x)=x^{3}, g(x)=x^{3}+1 / x^{2}$$

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x^{2}}{x+1}$$
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