Exploring Rational Root Theorem and Descartes' Rule of Signs

What does the Rational Root Theorem help us determine in a polynomial equation?

• The maximum degree of the polynomial

• The derivative of the polynomial

• The possible rational zeros based on coefficients

• The graphical representation of the polynomial

According to Descartes' Rule of Signs, what does a change in sign in the polynomial coefficients indicate?

• A potential rational zero

• A possible inflection point

• A possible real zero

• A potential maximum or minimum

How many positive real zeros are possible if a polynomial's coefficients change sign three times?

• Two or zero

• Three or one

• Four or two

• One or none

What does a sign change in the coefficients of f(x) typically indicate?

• A purely imaginary zero

• A constant function

• An undefined function point

• A possible real zero

What is the first step in applying the Rational Root Theorem to a polynomial?

• Identify the degree of the polynomial

• List all possible rational zeros

• Graph the polynomial

• Perform synthetic division

What does the Rational Root Theorem use to determine possible zeros?

• Integrals of the polynomial

• Derivatives of the polynomial

• Factors of the constant term and the leading coefficient

• Leading coefficient and degree of the polynomial

In the context of polynomial functions, what does the term 'rational zero' refer to?

• A zero that is complex

• A zero that is an irrational number

• A zero that is a whole number

• A zero that can be expressed as a fraction

When applying the Rational Root Theorem, what must be true for a number to be considered a potential zero?

• It must be a prime number

• It must be an irrational number

• It must be a positive integer

• It must be a factor of the leading coefficient or constant term

What is confirmed if the remainder in synthetic division is zero?

• The polynomial has no real roots

• The leading coefficient is zero

• The polynomial is non-factorable

• The divisor is a factor of the dividend

How can Descartes' Rule of Signs assist in narrowing down the testing of potential zeros?

• By increasing the number of potential rational zeros

• By eliminating non-real zeros

• By confirming all possible zeros

• By reducing the number of tests for negative zeros