Problem 61 In each case, find a polynomial ... [FREE SOLUTION]

Chapter 10: Problem 61

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only one \(x\) -intercept at \((3,0)\) and crosses the \(x\) -axis there.

Short Answer

Expert verified

The polynomial function is \(f(x) = x - 3\).

Step by step solution

Identify the condition for the intercept

The polynomial has one x-intercept at \(x = 3\). The fact that it crosses the x-axis there suggests that the root is of odd multiplicity. A simple choice is a root of multiplicity one.

Construct the factor

The factor corresponding to an x-intercept at \(x = 3\) is \( (x - 3) \).

Form the polynomial

The simplest polynomial with the factor \(x - 3\) is \(f(x) = (x - 3)\).

Verify the polynomial

Check that \(f(3) = 0\), confirming that it has the correct x-intercept. \ f(3) = 3 - 3 = 0 \.

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Key Concepts

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

x-intercepts

The x-intercept of a polynomial function is a point where the graph of the polynomial crosses or touches the x-axis. This happens when the y-value of the function is zero. For instance, if you have the point (3, 0), it means the graph intersects the x-axis at x = 3. Identifying x-intercepts is crucial because they provide insights into the behavior of the graph. In our given exercise, the x-intercept is at (3, 0), meaning the polynomial must satisfy f(3) = 0. This leads us to form a factor which corresponds to this intercept.

roots of polynomials

Roots of polynomials, also known as zeros, are the values of x for which the polynomial equation equals zero. These roots determine where the graph of the polynomial crosses or touches the x-axis. For example, if we have a polynomial function f(x), and f(3) = 0, then 3 is a root. In the exercise, the root was x = 3, leading to the factor (x - 3). The process of finding these roots involves solving equations. Finding roots gives us critical points to plot and understand the polynomial's behavior on a graph.

multiplicity of roots

Multiplicity of roots indicates the number of times a particular root is repeated in a polynomial. If a root has a multiplicity of one, it suggests that the graph will cross the x-axis at that point. On the other hand, if a root has a higher multiplicity (like 2, 3, etc.), it either touches without crossing or crosses with curvature due to higher derivatives. In our example, the x-intercept (3, 0) suggests an odd multiplicity鈥攕pecifically one鈥攕o it implies the simplest factor is (x - 3). Understanding multiplicity helps in sketching smoother and accurate graphs of polynomials.

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91影视

In each case, find a polynomial function whose graph behaves in the required manner. Answers may vary. The graph has only one \(x\) -intercept at \((3,0)\) and crosses the \(x\) -axis there.

Short Answer

Step by step solution

Identify the condition for the intercept

Construct the factor

Form the polynomial

Verify the polynomial

Key Concepts

x-intercepts

roots of polynomials

multiplicity of roots

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