RAICES RACIONALES DE UN POLINOMIO

Analyzing Rational Roots of a Polynomial

Understanding Rational Numbers

• A rational number is defined as any number x in the form a/b , where a and b are integers, and b neq 0 . This definition establishes the foundation for identifying rational roots in polynomials.

• For example, the decimal 0.05 can be expressed as 5/100 , which simplifies to 1/20 . Both forms confirm that it is a rational number since both numerator and denominator are integers.

The Rational Root Theorem

• The theorem states that if a polynomial P(x) = a_n x^n + ... + a_0 has integer coefficients (with a_n neq 0 and a_0 neq 0), then any rational root in the form of p/q must have:

• ** p** as a divisor of the constant term (a_0)

• ** q** as a divisor of the leading coefficient (a_n)

Example Polynomial Analysis

• Consider the polynomial equation:

• 9x^4 - 42x^3 + 3x^2 + 84x + 36 = 0. This polynomial is degree four, indicating it should have four roots, which could be real or complex.

• To find potential integer roots, we first identify divisors of the constant term (36). These divisors will help us determine possible integer solutions for this polynomial.

Finding Divisors

• The divisors of 36 include: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, and ±36. Each divisor needs to be tested to see if substituting it into the polynomial yields zero (indicating it's a root).

• For instance:

• Testing with -1 gives P(-1)=16 (not zero).

• Testing with other values like P(1)=100 also does not yield zero.

• However, testing P(3)=0 confirms that x = 3 is indeed an integer root. Thus far only integer roots have been identified.

Identifying Possible Rational Roots

• After confirming an integer root (x = 3), we now seek possible rational roots using:

• Divisors of constant term (36): ±1, ±2,...±36.

• Divisors of leading coefficient (9): ±1, ±3,...±9.

• We will create fractions from these divisors to explore potential rational roots in the form of p/q where p comes from divisors of constant term and q from those of leading coefficient.

Polynomial Roots and Rational Solutions

Exploring Possible Rational Roots

• The speaker discusses the calculation of potential fractional roots for a polynomial, noting that 1 is not a root despite being an integer.

• The evaluation of other integers like ±3 and -4 is mentioned, indicating that these should be checked as possible roots in future steps.

• The process of checking each potential rational root one by one is emphasized, highlighting the tedious nature of manual calculations.

Utilizing Technology for Root Verification

• The speaker suggests using GeoGebra to simplify the verification process for fractional roots instead of calculating manually.

• Initial evaluations show that p(1/3) does not yield zero; however, p(-2/3) results in zero, confirming it as a valid root.

Confirming Polynomial Roots

• It is confirmed that x = -2/3 is indeed a root of the polynomial p(x), which can be expressed as 9x^4 - 42x^3 + 84x + 36 = 0.

• Detailed calculations are provided to demonstrate how substituting x = -2/3 into the polynomial yields zero, validating its status as a rational solution.

Factoring the Polynomial

• The conclusion drawn from finding roots leads to writing the polynomial in factored form: p(x) = (x - 3)(x + 2/3).