Problem 44 - 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. $$p(x)=x^{4}+6 x^{3}-7 x^{2}+2 x-1$$

Step-by-Step Solution

Answer

The given polynomial could have 2 or 0 positive zeros and 1 or 0 negative zeros according to Descartes' Rule of Signs.

Step 1: Identify the given polynomial

The given polynomial is \(p(x) = x^4 + 6x^3 - 7x^2 + 2x - 1\).

Step 2: Apply Descartes' rule for positive zeros

Descending from the highest power of \(x\), list the signs of the coefficients. In this case, the signs are +, +, -, +, -. The number of sign changes is the potential number of positive real zeros. We have two sign changes in this case, so there could be 2 or 0 positive real zeros.

Step 3: Apply Descartes' rule for negative zeros

To find the potential number of negative real zeros, substitute \(x\) with \(-x\) for the given equation. This gives \(p(-x) = -x^4 + 6x^3 + 7x^2 + 2x - 1\). Again, descending from the highest power of \(x\), list the signs of the coefficients. They come out to be -, +, +, +, -. There's only one sign change, so there could be 1 or 0 negative real zero.

Key Concepts

Polynomial Zeros

When learning about polynomial zeros, it's essential to understand that they are the values of x for which the polynomial equals zero; in other words, they are the roots or solutions of the polynomial equation. The polynomial given in our exercise is a quartic (fourth degree) polynomial, meaning it will have up to four real or complex zeros, in accordance with the Fundamental Theorem of Algebra. The zeros provide crucial information about where the graph of the polynomial intersects the x-axis, and can be real or complex numbers. It’s vital to not confuse the term 'zero' with the value of 0—'zeros' here refer to the x-values that make the polynomial equal to 0.

Sign Changes

The sign changes in a polynomial are the points at which the coefficients switch from positive to negative or vice versa when you list them in descending order by the power of x. Descartes' Rule of Signs uses these sign changes to predict the number of positive and negative real zeros of a polynomial. This rule provides us with an interesting insight: the actual number of positive or negative real zeros of the polynomial is always either equal to the number of sign changes or less than that number by an even integer. It’s a bit like being handed clues in a mathematical detective story—while we don't get the exact answers outright, Descartes' Rule gives us a narrowed-down range to work within.

Positive Real Zeros

To determine the potential count of positive real zeros of a polynomial, Descartes' Rule of Signs tells us to analyze the polynomial as-is. The number of times the coefficient signs change is directly related to the number of positive real zeros the polynomial might have. For instance, in our example, the polynomial has two sign changes, indicating there could be 2 or 0 positive real zeros—since any reduction must be by an even number. It's a helpful way to set expectations before attempting more complex methods of finding the actual zeros, and provides guidance while sketching the behavior of the polynomial's graph.

Negative Real Zeros

For negative real zeros, Descartes' Rule instructs us to substitute every instance of x with -x and then determine the number of sign changes. It's like looking into a mirror that only reflects the negatives. This approach flips the signs of the terms with odd powers. We followed this process in our exercise and found one sign change, indicating the possibility of having either 1 or 0 negative real zeros. Despite its simplicity, this transformation is powerful—it helps us anticipate not only how many but also the character of the zeros, whether they lie to the left or right of the y-axis on the graph.