Problem 66 - Precalculus Mathematics for Calculus

Question

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=2 x^{3}-x^{2}+4 x-7$$

Step-by-Step Solution

Answer

The polynomial can have 3 or 1 positive real zeros, and 0 negative real zeros. Total real zeros can be 3.

Step 1: Identify the signs of coefficients

Examine the polynomial \(P(x) = 2x^3 - x^2 + 4x - 7\) and note the signs of each term's coefficient: \(+, -, +, -\).

Step 2: Count sign changes for positive zeros

Count the number of sign changes from one term to the next. From \(+\) to \(-\), \(-\) to \(+\), and \(+\) to \(-\), there are three sign changes. By Descartes' Rule of Signs, there could be 3 or 1 positive real zeros.

Step 3: Find \(P(-x)\)

Substitute \(x\) with \(-x\) in the original polynomial to find \(P(-x) = 2(-x)^3 - (-x)^2 + 4(-x) - 7\). Simplify to get \(P(-x) = -2x^3 - x^2 - 4x - 7\).

Step 4: Count sign changes for negative zeros

Examine \(P(-x) = -2x^3 - x^2 - 4x - 7\) for sign changes. The coefficients are \(-, -, -, -\), showing zero sign changes. Therefore, there are 0 negative real zeros.

Step 5: Calculate total number of real zeros

Using the degree of the polynomial, which is 3, the total number of real or complex zeros must be 3. Since there could be 3 or 1 positive real zeros and 0 negative real zeros, the possible total number of real zeros is 3.

Key Concepts

Polynomial Zeros

To understand polynomial zeros, imagine solving a puzzle where each piece fits perfectly into the equation. A zero of a polynomial is a value for the variable that makes the polynomial equal to zero.
In simple terms, these zeros are the 'answers' we search for that solve the polynomial equation when set to zero. For a polynomial like \( P(x) = 2x^3 - x^2 + 4x - 7 \), we want to find the values of \( x \) that make the equation equal to zero.
Determining these zeros can be crucial because:

• They provide insights into the behavior of the polynomial's graph.
• They help in solving equations involving polynomials.
• They can be real or complex numbers, with the total count dictated by the polynomial's degree.

Every polynomial of degree \( n \) will have \( n \) roots, though some may be complex and not visible on a real number line.

Sign Changes

Sign changes in a polynomial are key to applying Descartes' Rule of Signs. They occur when the coefficients of consecutive terms in a polynomial expression differ in sign, for example, from positive to negative.
In our example \( P(x) = 2x^3 - x^2 + 4x - 7 \), we see the sign changes as:

• From \(+2x^3\) to \(-x^2\)
• From \(-x^2\) to \(+4x\)
• From \(+4x\) to \(-7\)

Each of these transitions counts as a sign change. According to Descartes' Rule of Signs, this tells us how many positive real zeros the polynomial might have. Specifically, the number of positive real zeros is equal to the number of sign changes or less by an even number. Hence, our polynomial may have 3 or 1 positive real zeros.

Real Zeros

Real zeros are the values at which the polynomial touches or crosses the x-axis on a graph. They are 'real' because they can be represented on the number line as opposed to complex numbers.
Examining real zeros in a polynomial is crucial as they reveal where the function equals zero within the realm of real numbers. Using Descartes' Rule of Signs, we assess both positive and negative real zeros. For the polynomial \( P(x) = 2x^3 - x^2 + 4x - 7 \):Substituting \( x \) with \( -x \) and calculating \( P(-x) \) helps us check for negative real zeros by looking for sign changes in this new expression. In this example, there are no sign changes in \( P(-x) = -2x^3 - x^2 - 4x - 7 \), suggesting 0 negative real zeros.
Thus, considering possible positive zeros (3 or 1), and zero negative zeros, this polynomial could have all its zeros as real, matching the polynomial's degree, which is 3.