Question

Using Descartes' Rule, determine the nature of the roots of \(3 x^{5}-4 x^{4}+3 x^{3}-9 x^{2}-x+1=0\)

Short answer

The number of positive real roots can be either 4, 2, or 0, and the number of negative real roots can be either 2 or 0 in the polynomial \(3x^5 - 4x^4 + 3x^3 - 9x^2 - x + 1 = 0.\) The remaining roots are complex. Descartes' Rule alone is insufficient to determine the exact nature of the roots for this particular polynomial.

Step-by-step solution

Analyze the given polynomial

The given polynomial is: \[3x^5 - 4x^4 + 3x^3 - 9x^2 - x + 1\]

Count the sign changes in the original polynomial

We will count how many times the signs change when reading the terms from left to right: \(3x^5 (-4x^4) (+ 3x^3) (- 9x^2) (- x) (+ 1\) There are 4 sign changes in the given polynomial.

Apply Descartes' Rule for positive roots

Descartes' Rule states that the number of positive real roots is equal to the number of sign changes, or less than that by a positive even integer. In this case, there are 4 sign changes, so the number of positive real roots can be either 4, 2, or 0.

Replace x with -x in the polynomial

To find the number of negative roots, we first replace x with -x in the polynomial: \[3(-x)^5 - 4(-x)^4 + 3(-x)^3 - 9(-x)^2 -(-x) + 1\]

Simplify the polynomial with -x

After replacing x with -x and simplifying, the polynomial becomes: \[-3x^5 - 4x^4 - 3x^3 - 9x^2 + x + 1\]

Count the sign changes in the polynomial with -x

We count the sign changes in the polynomial with -x just like we did earlier: \((-3x^5) (-4x^4) (- 3x^3) (- 9x^2) (+ x) (+ 1\) There are 2 sign changes in the simplified polynomial with -x.

Apply Descartes' Rule for negative roots

Descartes' Rule states that the number of negative real roots is equal to the number of sign changes in the polynomial with -x, or less than that by a positive even integer. In this case, there are 2 sign changes, so the number of negative real roots can be either 2 or 0.

Analyze the results

Given that the polynomial is of degree 5, there are 5 roots in total (including real and complex roots). Using Descartes' Rule, we can have either 4, 2, or 0 positive real roots and either 2 or 0 negative real roots. So, the possible combinations of the nature of roots are: 1. 4 positive real roots, 1 negative real root (no complex roots) 2. 2 positive real roots, 2 negative real roots, 1 complex root (conjugate pair) 3. 0 positive real roots, 2 negative real roots, 3 complex roots (complex conjugate pairs) 4. 2 positive real roots, 0 negative real roots, 3 complex roots (complex conjugate pairs) Unfortunately, Descartes' Rule alone isn't enough to determine the exact nature of the roots for this particular polynomial. But, we can conclude that there are either 2 positive, 4 positive, or no positive real roots, and either 2 negative or no negative real roots with the remaining roots being complex.

Key concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, connected by operations of addition, subtraction, and multiplication. They can have one or more terms, such as 3x or 7x^4. A simple polynomial is just one term, like . The terms are combined with operators like addition or subtraction, for example:

• 2x + 3
• 4x^2 - 5x + 6

Polynomials are often categorized by their degree, which is the highest power of the variable in the expression. For example, in the polynomial 4x^3 + x^2 - 7, the degree is 3.
Understanding polynomials is essential to employing Descartes' Rule of Signs, as this rule helps us understand the number and type of solutions a polynomial equation may have.

Real Roots

Real roots of a polynomial are the solutions that are real numbers. They are the x-values where the polynomial equals zero, and they appear as the points where the graph of the polynomial crosses or touches the x-axis.
Finding real roots is important for understanding the graph and behavior of polynomials. Various techniques can be used to find them, including factoring, using the quadratic formula, or graphing the equation.

• Real roots are synonymous with x-intercepts.
• They can be rational (e.g., fractions) or irrational (e.g., square roots of numbers) values.

Descartes' Rule of Signs provides a method to estimate how many such real roots there may be for a polynomial.

Complex Roots

Complex roots involve numbers that have both a real part and an imaginary part. Imaginary numbers are multiples of the square root of -1 (denoted as i).
For example, a complex number might look like 3 + 4i.

• Complex roots in polynomials typically come in conjugate pairs, such as 3 + 4i and 3 - 4i.
• They do not appear on the standard x-y plane but can be visualized in the complex plane.

Understanding where complex roots come from is crucial for analyzing polynomials, especially those with real coefficients. Descartes' Rule of Signs helps identify the potential presence of complex roots when real roots don't account for all possible solutions.

Degree of a Polynomial

The degree of a polynomial is a major factor in determining the number of potential roots of the equation. It is defined as the highest power of the variable contained within the polynomial.
For instance, in the polynomial 7x^5 + 3x^3 - x + 2, the degree is 5. The degree not only dictates various mathematical properties of polynomials but also indicates how many roots (real and complex) it can have.

• A degree n polynomial will have exactly n roots in the complex number system.
• The degree helps use Descartes' Rule of Signs effectively to determine potential root scenarios.

When applying Descartes’ Rule of Signs, understanding the degree of the polynomial allows for accurate prediction of root combinations, as shown in the given solution with five possible roots.