Problem 36 - Algebra for College Students

Question

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$2 x^{3}-3 x^{2}+5 x-6=0$$

Step-by-Step Solution

Answer

The polynomial can have up to 3 roots: either 3 real roots, or 1 real root and 2 complex conjugate roots.

Step 1 - Determine the Degree of the Polynomial

The degree of the polynomial is the highest exponent of the variable x. In this equation, the highest exponent is 3, so the degree of the polynomial is 3.

Step 2 - Identify the Number of Roots

A polynomial of degree 3 can have at most 3 roots. These roots can be real or complex.

Step 3 - Compare Possible Root Types

Since the polynomial is of degree 3, it can have different combinations of roots: 3 real roots, 1 real root and 2 complex conjugate roots, or all 2 and 1 root (usually 1 real root, when the other roots are complex).

Step 4 - Use Descartes' Rule of Signs

By Descartes' Rule of Signs, count the number of sign changes in the polynomial. For the given polynomial, the sign changes in the coefficients are: +, -, +, -, which indicates: up to 3 possible positive roots, 1 positive root and 2 complex roots or just 1 positive root.

Key Concepts

degree of polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial equation. For instance, in the polynomial equation \(2x^{3} - 3x^{2} + 5x - 6 = 0\), the highest exponent present is 3.
Therefore, the degree of this polynomial is 3.
The degree of a polynomial dictates the maximum number of roots the equation can have.
In this case, since the degree is 3, it means the polynomial can have up to 3 roots.
These roots can be either real or complex.
Understanding the degree is crucial for predicting and analyzing the behavior of the polynomial.

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful method to predict the number of positive and negative real roots in a polynomial equation.
According to this rule, the number of positive real roots of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients, or it is less than that number by an even integer.
In the polynomial \(2x^{3} - 3x^{2} + 5x - 6 = 0\), observe the signs of the coefficients as follows: +, -, +, -.
Here, there are three changes in the signs:

• From + (coefficient of \(2x^{3}\)) to - (coefficient of \(-3x^{2}\))
• From - (coefficient of \(-3x^{2}\)) to + (coefficient of \(5x\))
• From + (coefficient of \(5x\)) to - (coefficient of \(-6\))

This indicates there could be up to three positive real roots. However, according to Descartes' rule, it is also possible to have one positive root, since any even decrease (3-2=1) is allowed as well.
Therefore, there could be 1 or 3 positive real roots in this polynomial.

real and complex roots

Roots of a polynomial can be either real or complex. Real roots are the solutions where the polynomial equals zero and can be plotted on a real number line.
Complex roots, on the other hand, include imaginary numbers and often come in conjugate pairs when the polynomial has real coefficients.

• A polynomial of degree 3, such as \(2x^{3} - 3x^{2} + 5x - 6 = 0\), can have different combinations of roots:
• 3 real roots
• 1 real root and 2 complex conjugate roots
• Or 1 real root when the other roots are complex

This means, in our given example, we can have a mix of 3 real roots or a combination of real and complex roots.
Complex conjugate roots appear as pairs like \(a+bi\) and \(a-bi\), where \(a\) and \(b\) are real numbers.
Understanding if a polynomial has real or complex roots can provide insights into the nature of solutions and their graphical representation.