Problem 39 - College Algebra

Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=3 x^{3}-10 x+9 ; \text { between }-3 \text { and }-2 $$

Step-by-Step Solution

Answer

By the Intermediate Value Theorem, the polynomial \(f(x)=3x^3-10x+9\) has a real zero between -3 and -2.

Step 1: Evaluate the Function at the Given Intervals

Evaluate the function at -3 and -2:\n\(f(-3) = 3*(-3)^{3} -10*(-3) + 9= -6\)\n\(f(-2) = 3*(-2)^{3} -10*(-2) + 9= 8\)

Step 2: Check the Sign Change

Notice the outputs, f(-3) and f(-2), are of different signs. This means that the function goes from negative to positive as x increases from -3 to -2.

Step 3: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem, since f is a polynomial (thus continuous everywhere), and since \(f(-3)<0\) and \(f(-2)>0\), there must be a number 'c' in the interval (-3,-2) such that \(f(c)=0\). In other words, there is a real zero between -3 and -2.

Key Concepts

Real Zero

In the context of polynomial functions, a *real zero* is a value of \(x\) for which the polynomial equals zero. This means that the polynomial "touches" or "crosses" the x-axis at this point. Finding real zeros is crucial because they inform us about the roots of the function, which is the point where the output of the function is zero. For example, consider a polynomial function\(f(x)\). If \(f(c) = 0\), where \(c\) is a real number, then \(c\) is a real zero of \(f(x)\).

Real zeros can be identified using various methods, including graphing, factoring, and by using the Intermediate Value Theorem in certain cases.

• Graphing helps to visually inspect where the function intersects the x-axis.
• Factoring involves breaking down the polynomial into simpler pieces to solve for zeros directly.
• The Intermediate Value Theorem is particularly useful when other methods are challenging or impossible.

Polynomial Function

A polynomial function is a type of mathematical expression that consists of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents.
A general polynomial function is expressed in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where:

• \(a_n, a_{n-1}, ..., a_1, a_0\) are constants known as coefficients,
• and \(n\) is a non-negative integer representing the degree of the polynomial.

Polynomial functions are essential because they can model a wide range of real-world phenomena, from physics to economics. They possess many useful properties, such as being smooth and continuous, which means there are no breaks or gaps in their graphs. This characteristic allows us to reliably use the Intermediate Value Theorem when analyzing potential real zeros within an interval.

Continuous Function

A continuous function is a function that has no holes, jumps, or breaks in its domain. In simple terms, you can draw the graph of the function without lifting your pencil from the paper.
For a function to be considered continuous at a point \(x = c\), it must satisfy three main conditions:

• The function \(f(x)\) is defined at \(x = c\).
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\).

Polynomial functions are a straightforward example of continuous functions due to their smooth nature when graphed. Thus, they have no abrupt changes in value, making them ideal candidates for applying the Intermediate Value Theorem. This theorem confirms that if a continuous function takes on both negative and positive values within an interval, there must exist at least one real zero within this interval, demonstrating the basic principle of how continuous functions behave along their graphs.