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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 35
Chapter 4, Problem 35

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x3−3x2+3x−1

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Write down the given polynomial function: \(f(x) = 5x^{3} - 3x^{2} + 3x - 1\).
To find the possible number of positive real zeros, count the number of sign changes in the coefficients of \(f(x)\). The coefficients are \(5\), \(-3\), \(3\), and \(-1\).
Count the sign changes: from \(5\) to \(-3\) (change), \(-3\) to \(3\) (change), and \(3\) to \(-1\) (change). So, there are 3 sign changes, meaning there can be 3, 1, or 0 positive real zeros (subtracting multiples of 2).
To find the possible number of negative real zeros, evaluate \(f(-x)\) by substituting \(-x\) into the function: \(f(-x) = 5(-x)^{3} - 3(-x)^{2} + 3(-x) - 1\).
Simplify \(f(-x)\) and count the sign changes in its coefficients to determine the possible number of negative real zeros, again considering that the number of negative zeros is the number of sign changes or less by multiples of 2.

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Key Concepts

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

Descartes's Rule of Signs

Descartes's Rule of Signs is a method used to determine the possible number of positive and negative real zeros of a polynomial function by counting the sign changes in the coefficients. The number of positive real zeros is equal to the number of sign changes in f(x) or less than that by an even number. For negative zeros, the rule is applied to f(-x).
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Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x that make the function equal to zero. Understanding the degree and behavior of polynomials helps in predicting the number and nature of their roots.
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Evaluating f(-x) to Find Negative Zeros

To apply Descartes's Rule of Signs for negative zeros, substitute -x into the polynomial to get f(-x). Then count the sign changes in the coefficients of f(-x). This process helps determine the possible number of negative real zeros by analyzing how the signs alternate after substitution.
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