Problem 48 - 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. $$p(x)=3 x^{4}-2 x^{3}+3 x^{2}-4 x+1$$

Step-by-Step Solution

Answer

The polynomial \(p(x) = 3x^4 - 2x^3 + 3x^2 - 4x + 1\) has either 4 or 0 positive real zeros and 0 negative real zeros.

Step 1: Identify the Coefficients

The coefficients of the polynomial \(p(x) = 3x^4 - 2x^3 + 3x^2 - 4x + 1\) are 3, -2, 3, -4, and 1.

Step 2: Apply Descartes' Rule for Positive Roots

Descartes' Rule of Signs states that the number of positive real zeros of a polynomial function corresponds to the number of sign changes between coefficients. For the given polynomial, the signs of the coefficients are +, -, +, -, +. There are therefore 4 changes in sign, so the polynomial has 4 or 4 - 2*2 = 0 positive real zeros.

Step 3: Apply Descartes' Rule for Negative Roots

To find the possible number of negative real roots, replace \(x\) with \(-x\) and observe for sign changes. The polynomial becomes \(p(-x) = 3x^4 + 2x^3 + 3x^2 + 4x + 1\), with coefficients 3, 2, 3, 4, and 1. All the coefficients are positive, and thus there are no sign changes. Hence, the polynomial has 0 negative real zeros.

Key Concepts

Polynomial Zeros

Understanding polynomial zeros is fundamental in algebra. A zero of a polynomial, sometimes referred to as a root, is a value of the variable that makes the polynomial equal zero. In simpler terms, if you plug this number into the polynomial in place of the variable, the result will be zero. It is at these points that the graph of the polynomial touches or crosses the x-axis.

For the polynomial \( p(x) = 3x^4 - 2x^3 + 3x^2 - 4x + 1 \), the zeros are those values of \( x \) which satisfy \( p(x) = 0 \). Identifying zeros is crucial for solving equations, graph plotting, and understanding the behavior of functions.

• The zeros can be real or complex.
• Real zeros imply that the graph of the polynomial touches the x-axis at these points.
• Complex zeros don't intersect the x-axis in the real coordinate plane.

The process of finding zeros can be cumbersome, but Descartes' Rule of Signs provides a useful shortcut for quantifying potential positive and negative zeros without finding the exact values.

Positive Zeros

Positive zeros, or positive roots, of a polynomial are achieved when the polynomial evaluates to zero with positive values of \( x \). Tracking these can be simplified using Descartes' Rule of Signs. The rule states that the number of positive real zeros of a polynomial function equals the number of times the sign of the coefficients changes in sequence, or less by a multiple of two.

In the polynomial \( p(x) = 3x^4 - 2x^3 + 3x^2 - 4x + 1 \), we observe the sequence of signs as +, -, +, -, +. Here, we notice four sign changes:

• From + (3) to - (-2)
• From - (-2) to + (3)
• From + (3) to - (-4)
• From - (-4) to + (1)

This means the polynomial can have either 4 positive zeros or fewer by multiples of 2 (for instance, 2 or 0). Hence, up to 4 positive zeros are possible, but through adjustments, this could also mean fewer, adhering to the multiple of two rule.

Negative Zeros

Negative zeros refer to the roots of the polynomial where the inputs are negative values of \( x \). To determine possible negative zeros using Descartes' Rule of Signs, you substitute \( x \) with \( -x \) and check the sequence of sign changes in the polynomial's coefficients.

For \( p(-x) \), if we transform the given polynomial \( p(x) = 3x^4 - 2x^3 + 3x^2 - 4x + 1 \) by substituting \( -x \), it changes to \( p(-x) = 3x^4 + 2x^3 + 3x^2 + 4x + 1 \).

Examining the signs of this polynomial's coefficients, we have: + (3), + (2), + (3), + (4), + (1). As we see, there are no sign changes, hence Descartes' Rule concludes there are 0 negative real zeros for this polynomial. This application simplifies the determination process and assists in predicting the nature of zeros based on sign consistency.