Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 18

Chapter 4, Problem 18

Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.

Verified step by step guidance

Step 1: Understand the problem. We need to show that the function $ f(x) = x^3 - 2x - 1 $ has a real zero between $ x = 1 $ and $ x = 2 $. This means we need to find values of $ f(x) $ at these points and check for a sign change.

Step 2: Calculate $ f(1) $. Substitute $ x = 1 $ into the function: $ f(1) = 1^3 - 2(1) - 1 $. Simplify this expression to find the value of $ f(1) $.

Step 3: Calculate $ f(2) $. Substitute $ x = 2 $ into the function: $ f(2) = 2^3 - 2(2) - 1 $. Simplify this expression to find the value of $ f(2) $.

Step 4: Analyze the results. Check the signs of $ f(1) $ and $ f(2) $. If $ f(1) $ and $ f(2) $ have opposite signs, then by the Intermediate Value Theorem, there is at least one real zero between $ x = 1 $ and $ x = 2 $.

Step 5: Conclude. If a sign change is observed, conclude that $ f(x) = x^3 - 2x - 1 $ has a real zero between $ x = 1 $ and $ x = 2 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, then it must take on every value between those two points at least once. This theorem is essential for proving the existence of a real zero in the given interval, as it guarantees that if the function changes signs between two points, a root exists in that interval.

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output, meaning there are no breaks, jumps, or holes in the graph. The function f(x) = x^3 - 2x - 1 is a polynomial, and all polynomial functions are continuous, which is a critical property for applying the Intermediate Value Theorem.

Evaluating Function Values

To apply the Intermediate Value Theorem, we need to evaluate the function at the endpoints of the interval. By calculating f(1) and f(2), we can determine if the function changes sign between these two points, which indicates the presence of a real zero. This step is crucial for establishing the conditions required by the theorem.

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Related Practice

Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

$f (x) = (x - 3)^{2}$

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Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. $f (x) = x^{3} - x^{2} - 9 x + 9$

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 4x²+ 1 ≥ 4x

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. $5 x is less than or equal to 2 minus 3 x squared$

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Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the difference between y and w.

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. $x squared minus 4 x is greater than or equal to 0$

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