descartes rule of signs to determine the possible number of positive and negative real zeros of: p(x)=-x^4+3x^3+2x^2-10x+12

Answer 1

DeCartes Rule of signs provides the possible number of positive and negative real roots to a polynomial function.

For the positive roots, count the number of sign changes within the expression p(x)

p(x) = -x⁴ + 3x³ + 2x² - 10x + 12

In this expression there are 3 sign changes: -x⁴ to +3x³ (change 1), +2x² to -10x (change 2), and -10x to +12 (change 3).

So, there are either 3 or 1 positive real roots.

For the negative roots, count the number of sign changes within the expression p(-x)

p(-x) = -(-x)⁴ + 3(-x)³ + 2(-x)² - 10(-x) + 12 = -x⁴ - 3x³ + 2x² + 10x + 12

in this expression, there is 1 sign change: -3x³ to +2x²

So, there is 1 negative real root