Problem 49 - 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. $$p(x)=x^{5}+3 x^{4}-4 x^{2}+10$$

Step-by-Step Solution

Answer

The polynomial \(p(x) = x^{5} +3x^{4} - 4x^{2} + 10\) has either 2 or 0 positive real roots and 1 or 0 negative real roots.

Step 1: Analyzing the polynomial to find signs of terms.

Firstly, take the polynomial \(p(x) = x^{5} +3x^{4} - 4x^{2} + 10\). Notice that the signs of the coefficients change twice. It switches from positive to positive (from \(x^{5}\) to \(3x^{4}\)) then to negative (from \(3x^{4}\) to \(-4x^{2}\)) and finally, back to positive (from \(-4x^{2}\) to \(10\)). Thus, there are exactly two changes in sign.

Step 2: Applying Descartes' Rule of Signs

According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less than that by an even integer. Hence, the given function \(p(x)\) has either 2 positive real roots or none (2-2=0, an even number).

Step 3: Finding the number of negative roots

Now, replace \(x\) with \(-x\) in the equation, resulting in the new function \(p(-x)\). Thus, we get \(p(-x) = -x^{5} + 3x^{4} + 4x^{2} + 10\). We can see that the signs change once in this polynomial. Again according to Descartes' Rule of Signs, the number of negative roots of the polynomial is either the number of sign changes or less than that by an even number. Hence, the function \(p(x)\) has either 1 negative real root or none (1-2=-1, but we cannot have a negative number of roots, so we only consider 1-0=1).

Key Concepts

Polynomial Roots

Polynomial roots, also known as zeros, are the values for which a polynomial equals zero. These roots can be real or complex numbers. In the context of a polynomial like \(p(x) = x^5 + 3x^4 - 4x^2 + 10\), the task is to determine where this polynomial crosses the x-axis, meaning where \(p(x) = 0\).
To find these roots, one usually employs methods such as factoring the polynomial, using the quadratic formula (for quadratic polynomials), or applying numerical methods. However, with Descartes' Rule of Signs, we're focusing specifically on the real roots—the values of \(x\) resulting in \(p(x) = 0\).
It's important to understand that not all polynomials have real roots. Sometimes, the roots may be complex and thus not visible on the real number line. Descartes' rule assists us in figuring out the potential number of positive and negative real roots without necessarily solving for these roots directly.

Sign Changes in Polynomials

Sign changes in a polynomial refer to alterations in the sign of its coefficients as you move from one term to the next within the polynomial. To use Descartes' Rule of Signs effectively, identifying these sign changes is crucial.
Take the polynomial \(p(x) = x^{5} + 3x^{4} - 4x^{2} + 10\). As we analyze it, we note its terms as follows:
\(x^{5}\) is positive, \(3x^{4}\) is positive, \(-4x^{2}\) is negative, and finally, \(+10\) is positive.
From here, we observe that the sign changes occur twice:

• From the positive \(3x^{4}\) to the negative \(-4x^{2}\)
• From the negative \(-4x^{2}\) to the positive constant term \(+10\)

This indicates two sign changes, a critical input for applying Descartes’ Rule of Signs. Each change in sign represents a potential positive root.

Positive and Negative Roots

Descartes' Rule of Signs provides a systematic way to determine the number of positive and negative roots of a polynomial.

First, for positive roots, you examine the original polynomial. The number of positive real roots equals the number of sign changes in the polynomial's sequence of coefficients, or less than that by an even number. In our example \(p(x) = x^{5} + 3x^{4} - 4x^{2} + 10\), there are two sign changes. Therefore, it has 2 or 0 positive roots.

To find the number of negative real roots, substitute \(x\) with \(-x\) in the polynomial to create \(p(-x)\). Perform the same sign analysis on \(p(-x) = -x^{5} + 3x^{4} + 4x^{2} + 10\):

• \(-x^{5}\) to \(+3x^{4}\) is a sign change

This single sign change implies there is one negative real root or zero, depending on further subtraction by even numbers.

So, Descartes’ Rule informs us that for the given polynomial, we expect either 2 positive roots and 1 negative root, or no positive roots and 1 or no negative roots.