Problem 37 - Algebra and Trigonometry

Question

In Exercises 33–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=2 x^{4}-5 x^{3}-x^{2}-6 x+4 $$

Step-by-Step Solution

Answer

The given function may have either 2 or 0 positive real zeros and 3 or 1 negative real zeros.

Step 1: Count Sign Changes in f(x)

Firstly, arrange the terms of the given polynomial \( f(x) = 2 x^{4} - 5 x^{3} - x^{2} - 6 x + 4 \) in descending powers of x and count the number of sign changes: (+,-,-,-,+) which gives us 2. Therefore, we can have 2 or 2-2*1=0 positive real zeros.

Step 2: Count Sign Changes in f(-x)

Next, substitute -x for x in the equation to determine the number of negative real zeros. The function becomes \( f(-x) = 2(-x)^{4} + 5(-x)^{3} - (-x)^{2} + 6(-x) + 4 = 2 x^{4} + 5 x^{3} - x^{2} - 6 x + 4 \) Counting the sign changes (+,+,+,-,+) results in 3 changes. Hence, we could have 3 or 3-2*1=1 negative real zeros.

Step 3: Combine Results

Combine the results from steps 1 and 2. Therefore, the function \( f(x) = 2 x^{4} - 5 x^{3} - x^{2} - 6 x + 4 \) can possibly have either 2 or 0 positive real zeros and either 3 or 1 negative real zeros as per Descartes’ Rule of Signs.

Key Concepts

Polynomial Functions

Polynomial functions consist of terms that involve variables raised to positive integer powers. Each term is a monomial, and in the polynomial, they are usually arranged in descending order of their exponents. The general form of a polynomial function can be written as:
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0 \]
where:

• \( a_n, a_{n-1}, \text{...}, a_1, \text{and} \ a_0 \) are constants (coefficients)
• \( x \) is the variable
• \( n \) is the degree of the polynomial

Polynomial functions are foundational in algebra and calculus because they allow for modeling a diverse range of behaviors, due to their smooth and continuous nature. Understanding their structure helps us identify and predict the behavior of polynomial curves, especially when analyzing their roots and extrema.

Real Zeros

Real zeros of a polynomial function are the values of \( x \) for which the function equals zero. In other words, they are the points where the graph of the polynomial crosses or touches the x-axis. To find real zeros, we solve the equation \( f(x) = 0 \).
Real zeros give us critical insight into the nature of a polynomial function. They represent solutions that can be graphed on the Cartesian plane, as opposed to imaginary zeros which do not appear on a standard graph. Finding real zeros is a primary objective when analyzing and interpreting polynomial equations.

Sign Changes

Sign changes in a polynomial function hold the key to estimating the number of real zeros. Descartes's Rule of Signs utilizes these variations in sign to predict how many positive or negative real zeros the polynomial may possess.
When analyzing for positive real zeros, observe the polynomial in its original form, listing terms and noting where signs change from positive to negative or vice versa. For instance, examining \( f(x) = 2x^4 - 5x^3 - x^2 - 6x + 4 \) reveals two sign changes: (+ to -, and - to +).
For determining negative real zeros, substitute \( -x \) for \( x \) in the polynomial and count sign changes in the resulting expression. In the case of the given polynomial, the changes amount to three, aiding our estimate for negative real zeros.

Positive and Negative Zeros

Positive and negative zeros of polynomial functions correlate directly with sign changes deduced through Descartes's Rule of Signs. Positive zeros are potential solutions obtained from the original polynomial equation, while negative zeros result from substituting \( -x \) into the function.
By applying Descartes's Rule of Signs:

• Two sign changes in the polynomial \( f(x) = 2x^4 - 5x^3 - x^2 - 6x + 4 \) suggest there could be two or zero positive real zeros.
• Three sign changes in \( f(-x) \) suggest the possibilities of having three or one negative real zeros.

This rule provides potential counts, not exact numbers, guiding us on possibilities among real roots of the polynomial function, simplifying analysis by narrowing down potential solutions before rigorous calculations.