Problem 45 - Precalculus : Building Concepts and Connections

Question

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=-2 x^{3}+x^{2}-x+1$$

Step-by-Step Solution

Answer

The number of positive zeros of the given polynomial is 3 or 1 (since it must be less by an even number) and the number of negative zeros is 0.

Step 1: Identify the Polynomial

The polynomial given is \(p(x)=-2x^3+x^2-x+1\).

Step 2: Count the Number of Sign Changes for Positive Zero

For positive zero, keep the variable as \(x\). Now count the sign changes in the terms of the polynomial. After arranging it in decreasing order of power, \(p(x)=-2x^3+x^2-x+1\), the signs of the terms go from negative, to positive, then again to negative and finally to positive. Therefore, there are 3 changes in sign.

Step 3: Count the Number of Sign Changes for Negative Zero

For negative zero, substitute \(-x\) for \(x\) in the polynomial. The polynomial becomes \(p(-x)= -2(-x)^3+(-x)^2-(-x)+1= 2x^3+x^2+x+1\). Now count the sign changes. There's no sign change in the resulting polynomial, therefore, the number of negative roots is 0.

Key Concepts

Polynomial Roots

When solving polynomial equations, one of the main objectives is to find the roots. A polynomial root is a value of the variable that makes the polynomial equal to zero. For instance, if you have a polynomial \(p(x)\), and \(p(a) = 0\), then \(a\) is a root of the polynomial. These roots are crucial in various fields, including engineering, physics, and economics, as they can indicate points where certain values reach zero or change direction.

In algebra, the Fundamental Theorem of Algebra states that every non-zero single-variable polynomial with complex coefficients has as many roots as its degree, when counted with multiplicity. Hence, a polynomial of degree 3, like the one in our exercise, will generally have 3 roots. Some of these roots can be real numbers while others might be complex numbers. Understanding the nature and number of roots can give insight into the behavior and characteristics of the polynomial.

Positive Zeros in Polynomials

Positive zeros of a polynomial are the real roots that are greater than zero. To determine the number of such zeros, one useful method is Descartes' Rule of Signs, which provides a quick way to estimate the count by examining the signs of the polynomial's coefficients.

Based on Descartes' Rule of Signs, the number of positive real zeros of a polynomial corresponds to the number of sign changes between consecutive non-zero coefficients or is fewer by an even number. For example, consider the polynomial \(p(x)=-2x^3+x^2-x+1\).

• The coefficients are \(-2, +1, -1, +1\).
• The signs change from negative to positive (\(-2\) to \(+1\)), positive to negative (\(+1\) to \(-1\)), and negative to positive (\(-1\) to \(+1\)).

Thus, there are three sign changes, indicating that there could be 3, 1, or no positive roots. This ambiguity arises because we're only counting real roots and dismissing integer multiples in their count.

Sign Changes in Polynomials

Understanding sign changes in polynomials is vital for applying Descartes' Rule of Signs. To identify these changes, you must look carefully at the sequence of polynomial coefficients. A 'sign change' occurs when consecutive coefficients have different signs.

For example, in the polynomial \(p(x)=-2x^3 + x^2 - x + 1\), the sequence of signs is: negative (\(-2\)), positive (\(+1\)), negative (\(-1\)), then positive (\(+1\)) again. Sign changes are then recognized as follows:

• From \(-2\) to \(+1\) is a change from negative to positive.
• From \(+1\) to \(-1\) is a change from positive to negative.
• Finally, from \(-1\) to \(+1\) marks another change from negative to positive.

Each change suggests the potential for a positive root. Keep in mind that while counting these changes guides us toward the maximum number of positive roots, not all sign changes guarantee an actual root. The rule helps determine possibilities, and analyzing polynomial behavior further will refine these predictions.