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Chapter 3: Problem 40

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. \(f(x)=3 x^{3}-8 x^{2}+x+2 ;\) between 2 and 3

Short Answer

Expert verified
Yes, by the Intermediate Value Theorem there is a root between 2 and 3 for the given polynomial.

Step by step solution

01

Evaluate f(2) and f(3)

Let's substitute \(x = 2\) and \(x = 3\) in \(f(x)\):\n\nFor \(f(2)\), we get: \(f(2) = 3(2)^3 - 8(2)^2 + 2 + 2 = -4\)\n\nFor \(f(3)\), we get: \(f(3) = 3(3)^3 - 8(3)^2 + 3 + 2 = 10\)
02

Apply the Intermediate Value Theorem

According to the IVT, if \(0\) is within the interval between \(f(2)\) and \(f(3)\), then there should be a root between \(2\) and \(3\).\n\nAs we know, \(f(2) = -4\) and \(f(3) = 10\), and \(0\) is indeed between \(-4\) and \(10\).\n\nTherefore, by the IVT, there exists a real root between \(2\) and \(3\).

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Key Concepts

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

Real Zero of a Polynomial
Understanding the concept of a 'real zero of a polynomial' is fundamental in algebra and calculus. A real zero, also known as a root, is where the polynomial intersects the x-axis, meaning the output of the function, or f(x), is zero.

Let's consider the function from our exercise, \(f(x)=3x^3-8x^2+x+2\). We are looking for values of x where \(f(x)=0\). When we evaluate the function at the integers 2 and 3, we find \(f(2)=-4\) and \(f(3)=10\), indicating that the function values have different signs. The Intermediate Value Theorem assures us that since \(f(x)\) changes from negative to positive between these two x-values, there must be at least one real zero between them.
Evaluating Polynomials
When 'evaluating polynomials', we simply substitute the value of x into the polynomial and calculate the result. This process is a direct application of arithmetic operations involving power, multiplication, and addition.

Let's illustrate this using our provided polynomial \(f(x)\). To evaluate \(f(2)\), we substitute 2 into the polynomial, yielding \(3(2)^3 - 8(2)^2 + 2 + 2\), which simplifies to -4. Similarly, calculating \(f(3)\) involves substituting 3 into the polynomial, resulting in \(3(3)^3 - 8(3)^2 + 3 + 2\), which equals 10. These evaluations are crucial steps in applying the Intermediate Value Theorem, as they establish the necessary interval where a zero exists.
Mathematical Proofs
The process of forming 'mathematical proofs' involves creating a logical and convincing argument that a particular statement is true. Proofs can take many forms, such as direct, by contradiction, or by induction. The Intermediate Value Theorem is often used as a foundation for proving the existence of roots.

In the context of our exercise, the theorem helps us prove the existence of a real zero for a polynomial function within a specific interval by showing that the function values at the endpoints have opposite signs. By following this logical progression from evaluation to application of the theorem, we not only find that a zero must exist, but we also implicitly engage in the foundational practice of mathematical proof.

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Most popular questions from this chapter

Use the four-step procedure for solving variation problems given on page 424 to solve. \(a\) varies directly as \(b\) and inversely as the square of \(c . a=7\) when \(b=9\) and \(c=6 .\) Find \(a\) when \(b=4\) and \(c=8\).

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics I has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2

Solve each inequality using a graphing utility. $$ 2 x^{2}+5 x-3 \leq 0 $$

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies jointly as \(y\) and the square of \(z\).

The functions $$f(x)=0.0875 x^{2}-0.4 x+66.6$$ and $$g(x)=0.0875 x^{2}+1.9 x+11.6$$ model a car's stopping distance, \(f(x)\) or \(g(x),\) in feet, traveling at \(x\) miles per hour. Function \(f\) models stopping distance on dry pavement and function g models stopping distance on wet pavement. The graphs of these functions are shown for \(\\{x | x \geq 30\\} .\) Notice that the figure does not specify which graph is the model for dry roads and which is the model for wet roads. Use this information to solve. (GRAPH CANNOT COPY). a. Use the given functions to find the stopping distance on dry pavement and the stopping distance on wet pavement for a car traveling at 55 miles per hour. Round to the nearest foot. b. Based on your answers to part (a), which rectangular coordinate graph shows stopping distances on dry pavement and which shows stopping distances on wet pavement? c. How well do your answers to part (a) model the actual stopping distances shown in Figure 3.43 on page \(411 ?\) d. Determine speeds on wet pavement requiring stopping distances that exceed the length of one and one-half football fields, or 540 feet. Round to the nearest mile per hour. How is this shown on the appropriate graph of the models?
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