Descartes’ Rule of Signs - Algebra-2

1. Fundamental Concepts

Definition: Descartes' Rule of Signs is a method used to determine the possible number of positive and negative real roots of a polynomial equation.
Positive Roots: The number of positive real roots is equal to the number of sign changes in the sequence of coefficients, or less than that by a multiple of 2.
Negative Roots: The number of negative real roots is equal to the number of sign changes in the sequence of coefficients of the polynomial with the variables replaced by their negatives, or less than that by a multiple of 2.

Sign Changes:

Number of positive roots = Number of sign changes in f(x) or less by a multiple of 2

Negative Roots:

Number of negative roots = Number of sign changes in f(-x) or less by a multiple of 2

Application: Used to narrow down the possible number of real roots, aiding in solving polynomial equations

Problem: Determine the possible number of positive and negative real roots for the polynomial $$f(x) = x^3 - 2x^2 - 5x + 6$$

Count the sign changes in $$f(x) = x^3 - 2x^2 - 5x + 6$$ :
- From $$+x^3$$ to $$-2x^2$$ : 1 change
- From $$-2x^2$$ to $$-5x$$ : 0 changes
- From $$-5x$$ to $$+6$$ : 1 change
Total sign changes: 2
Possible number of positive roots: 2 or 0
Count the sign changes in $$f(-x) = (-x)^3 - 2(-x)^2 - 5(-x) + 6 = -x^3 - 2x^2 + 5x + 6$$ :
- From $$-x^3$$ to $$-2x^2$$ : 0 changes
- From $$-2x^2$$ to $$+5x$$ : 1 change
- From $$+5x$$ to $$+6$$ : 0 changes
Total sign changes: 1
Possible number of negative roots: 1

Validation: The polynomial can be factored as $$(x-1)(x+2)(x-3)$$ , which confirms 2 positive roots and 1 negative root.

Problem: Determine the possible number of positive and negative real roots for the polynomial $$g(x) = 2x^4 - 3x^3 + x^2 - 7x + 8$$

Count the sign changes in $$g(x) = 2x^4 - 3x^3 + x^2 - 7x + 8$$ :
- From $$+2x^4$$ to $$-3x^3$$ : 1 change
- From $$-3x^3$$ to $$+x^2$$ : 1 change
- From $$+x^2$$ to $$-7x$$ : 1 change
- From $$-7x$$ to $$+8$$ : 1 change
Total sign changes: 4
Possible number of positive roots: 4, 2, or 0
Count the sign changes in $$g(-x) = 2(-x)^4 - 3(-x)^3 + (-x)^2 - 7(-x) + 8 = 2x^4 + 3x^3 + x^2 + 7x + 8$$ :
- No sign changes
Total sign changes: 0
Possible number of negative roots: 0

Validation: The polynomial does not factor easily, but the rule of signs indicates 4, 2, or 0 positive roots and 0 negative roots.

Systematic Counting: Carefully count the sign changes in the sequence of coefficients.
Substitution Method: Substitute $$-x$$ into the polynomial to find the number of negative roots.
Verification: Use synthetic division or other methods to verify the roots once the possible number is determined.