Problem 25
Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=5 x^{3}-3 x^{2}+3 x-1 $$
Step-by-Step Solution
Verified Answer
The function \(f(x) = 5x^3 - 3x^2 + 3x - 1\) may have 3 or 1 positive real zeros and 2 or 0 negative real zeros.
1Step 1: Counting sign changes in the function
The given function \(f(x) = 5x^3 - 3x^2 + 3x - 1\) has coefficients 5, -3, 3, -1. From positive to negative and vice versa, there are three sign changes. So, the number of positive real zeros in the function is either 3 or 3-2=1.
2Step 2: Determine function with reversed sign and count sign changes
Apply the rule to the function \(f(-x)\). Replace \(x\) with \(-x\) in the function. We get \(f(-x) = 5(-x)^3 - 3(-x)^2 + 3(-x) - 1 = -5x^3 - 3x^2 - 3x - 1\). Now, count the sign changes. There are two changes. Therefore, the number of negative real zeros is either 2 or 2-2=0.
3Step 3: Conclusion
So, according to Descartes' Rule of Signs, the function \(f(x) = 5x^3 - 3x^2 + 3x - 1\) may have 3 or 1 positive real zeros and 2 or 0 negative real zeros.
Key Concepts
Real ZerosSign ChangesPolynomial Functions
Real Zeros
When we talk about real zeros of a polynomial function, we're referring to the solutions of the equation where the function equals zero. Real zeros can be thought of as the x-values where the graph of the function crosses or touches the x-axis. For example, if you have a polynomial function like \( f(x) = 5x^3 - 3x^2 + 3x - 1 \), its real zeros are the x-values making the whole expression equal to zero.
These zeros are important because they give us points where the output of the function is zero — a valuable insight for understanding the behavior of the graph. However, determining these zeros analytically can sometimes be challenging. And this is where Descartes’s Rule of Signs comes in handy. It helps us determine how many real zeros exist without solving the equation directly.
Utilizing this rule involves counting the number of sign changes in the polynomial itself and its transformed version, offering a quick way to estimate the number of positive and negative real zeros.
These zeros are important because they give us points where the output of the function is zero — a valuable insight for understanding the behavior of the graph. However, determining these zeros analytically can sometimes be challenging. And this is where Descartes’s Rule of Signs comes in handy. It helps us determine how many real zeros exist without solving the equation directly.
Utilizing this rule involves counting the number of sign changes in the polynomial itself and its transformed version, offering a quick way to estimate the number of positive and negative real zeros.
Sign Changes
The concept of sign changes plays a crucial role in Descartes's Rule of Signs. Essentially, a sign change occurs when you move from one term to the next in a polynomial, and the coefficient changes from positive to negative or vice versa.
For the function \( f(x) = 5x^3 - 3x^2 + 3x - 1 \), we start with a positive coefficient (for \( x^3 \)) and move to a negative one (for \( x^2 \)), reflecting a sign change. Again, as we move from the negative \( x^2 \) term to the positive \( x \) term, there's another sign change. Finally, we move from the positive \( x \) term to the negative constant term \(-1\), resulting in a third sign change. In total, there are three sign changes, hinting that there could be three or one positive real zeros.
Similarly, when determining the possible number of negative real zeros, we modify the function by replacing \( x \) with \( -x \) and count the sign changes on this new form. This gives a different picture of when and how many negative real zeros may appear.
For the function \( f(x) = 5x^3 - 3x^2 + 3x - 1 \), we start with a positive coefficient (for \( x^3 \)) and move to a negative one (for \( x^2 \)), reflecting a sign change. Again, as we move from the negative \( x^2 \) term to the positive \( x \) term, there's another sign change. Finally, we move from the positive \( x \) term to the negative constant term \(-1\), resulting in a third sign change. In total, there are three sign changes, hinting that there could be three or one positive real zeros.
Similarly, when determining the possible number of negative real zeros, we modify the function by replacing \( x \) with \( -x \) and count the sign changes on this new form. This gives a different picture of when and how many negative real zeros may appear.
Polynomial Functions
Polynomial functions are an essential topic in algebra, representing a wide range of mathematical phenomena. They are equations that involve sums, differences, and constant non-negative integer powers of \( x \). For example, \( f(x) = 5x^3 - 3x^2 + 3x - 1 \) is a third-degree polynomial function.
These functions can exhibit diverse behaviors based on their coefficients and degree, which is the highest exponent present in the function. The degree gives a basic idea of the maximum number of real zeros a polynomial can have. In our example, the degree is 3, suggesting that, in theory, there could be up to three real zeros.
Exploring polynomial functions using strategies like Descartes's Rule of Signs enriches our understanding of their zero-discovery potential. This rule provides an accessible way to calculate the real zero estimates, which otherwise might seem daunting for high-degree polynomials. By grasping these concepts, you can unlock a deeper appreciation of the structure and solution dynamics of polynomials.
These functions can exhibit diverse behaviors based on their coefficients and degree, which is the highest exponent present in the function. The degree gives a basic idea of the maximum number of real zeros a polynomial can have. In our example, the degree is 3, suggesting that, in theory, there could be up to three real zeros.
Exploring polynomial functions using strategies like Descartes's Rule of Signs enriches our understanding of their zero-discovery potential. This rule provides an accessible way to calculate the real zero estimates, which otherwise might seem daunting for high-degree polynomials. By grasping these concepts, you can unlock a deeper appreciation of the structure and solution dynamics of polynomials.
Other exercises in this chapter
Problem 25
In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$
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In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that a
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Find the vertical asymptotes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x+4)}$$
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Divide using synthetic division. $$\left(x^{2}-5 x-5 x^{3}+x^{4}\right) \div(5+x)$$
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