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Chapter 3: Problem 106

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There is more than one third-degree polynomial function with the same three \(x\) -intercepts.

Short Answer

Expert verified
The statement is true. There can indeed be more than one third-degree polynomial function with the same x-intercepts.

Step by step solution

01

Understand the concept of x-intercepts

An x-intercept of a function is a point at which the value of the function is zero, i.e., it is a root of the function. In terms of a graph, these x-intercepts are the points where the graph crosses or touches the x-axis.
02

Recognize how polynomials work

A polynomial function of third degree generally looks like this: \(f(x) = a(x-b)(x-c)(x-d)\), where \(a\), \(b\), \(c\), and \(d\) are constants. In this case, \(b\), \(c\), and \(d\) are the x-intercepts of the function. Notice that the x-intercepts only depends on the constants \(b\), \(c\), and \(d\) and not on the constant \(a\). This means that changing \(a\) does not affect the x-intercepts of the function.
03

Evaluate the statement

Given that the constant \(a\) does not affect the x-intercepts of the function and that \(a\) can be any real number, it is correct to say that there is more than one third-degree polynomial function with the same three x-intercepts. Any changes to \(a\) would simply stretch or squeeze the graph vertically but would not change the x-intercepts.

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