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Chapter 3: Problem 44

Use Descartes's rule of signs to discuss the possibilities for the roots of each equation. Do not solve the equation. $$2 x^{3}-3 x^{2}+5 x-6=0$$

Short Answer

Expert verified
There are either 3 or 1 positive roots and no negative roots.

Step by step solution

01

Identify the Polynomial

Given the polynomial equation: 2x^3 - 3x^2 + 5x - 6 = 0
02

Evaluate Sign Changes for Positive Roots

Look at the signs of the coefficients in the polynomial: 2 (positive), -3 (negative), 5 (positive), and -6 (negative). By inspecting the changes in signs: - From +2 to -3 (1 change), - From -3 to +5 (2 changes), - From +5 to -6 (3 changes). Therefore, there are 3 sign changes.
03

Determine Possible Positive Roots

Using Descartes's rule of signs, the number of positive real roots is equal to the number of sign changes or less than that by an even number. Thus, the possibilities for the number of positive real roots are: 3, 1.
04

Evaluate Sign Changes for Negative Roots

To determine the number of possible negative roots, substitute x with -x and simplify: 2(-x)^3 - 3(-x)^2 + 5(-x) - 6 = 0 -2x^3 - 3x^2 - 5x - 6 = 0 Now, evaluate the sign changes: - From -2 to -3 (no change), - From -3 to -5 (no change), - From -5 to -6 (no change). Therefore, there are 0 sign changes.
05

Determine Possible Negative Roots

Using Descartes's rule of signs, the number of negative real roots is equal to the number of sign changes or less than that by an even number. Since there are 0 changes, there are no negative real roots.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial roots
When working with polynomial equations, like the one in the exercise (2x^3 - 3x^2 + 5x - 6 = 0), the 'roots' of the polynomial are the values of x that make the equation equal to zero. Understanding the roots of a polynomial helps in graphing and analyzing the equation.
There are various methods to find these roots, but one useful tool is Descartes's rule of signs. This rule helps in predicting the number of positive and negative real roots based on the signs of the coefficients in the polynomial.
positive real roots
Using Descartes's rule of signs, we analyze the changes in the signs of the coefficients of our polynomial to determine the number of positive real roots.
For the polynomial 2x^3 - 3x^2 + 5x - 6 = 0, observe the signs of the coefficients:
  • 2 (positive)

  • -3 (negative)

  • 5 (positive)

  • -6 (negative)
Count the sign changes:
  • From +2 to -3 (1 change)

  • From -3 to +5 (2 changes)

  • From +5 to -6 (3 changes)

This gives us 3 sign changes. According to Descartes's rule, the number of positive real roots is equal to the number of sign changes or less than that number by an even number. Therefore, our polynomial can have 3 or 1 positive real roots.
negative real roots
To find the number of negative real roots, we substitute x with -x in our polynomial equation and simplify:
2(-x)^3 - 3(-x)^2 + 5(-x) - 6 = 0
Simplifying, we get:
-2x^3 - 3x^2 - 5x - 6 = 0
Now, we check the sign changes in this new equation:
  • -2 (negative)

  • -3 (negative)

  • -5 (negative)

  • -6 (negative)
We see there are no sign changes.
Thus, Descartes's rule of signs tells us there are no negative real roots for this polynomial.
sign changes
The concept of 'sign changes' is fundamental in Descartes's rule of signs. A 'sign change' occurs when consecutive coefficients of the polynomial switch from positive to negative or negative to positive.
For example, in the polynomial 2x^3 - 3x^2 + 5x - 6 = 0, we observe:
  • +2 to -3 (positive to negative)

  • -3 to +5 (negative to positive)

  • +5 to -6 (positive to negative)
Counting these changes gives us 3 sign changes, which indicates the maximum number of positive real roots possible.
Conversely, when determining the number of negative real roots by substituting x with -x, we observed no sign changes in the derived polynomial -2x^3 - 3x^2 - 5x - 6 = 0, meaning there are no negative real roots.

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