Final answer:
To find the rational roots of the equation 3x^4 - 26x^3 + 81x^2 - 120x - 50 = 0 with -2 <= x <= 5, use the Rational Root Theorem by considering all possible factors of the constant term divided by factors of the leading coefficient. Substituting each value for x in the equation will determine which roots satisfy the equation.
Step-by-step explanation:
To find the rational roots of the equation 3x^4 - 26x^3 + 81x^2 - 120x - 50 = 0 with -2 ≤ x ≤ 5, we can use the Rational Root Theorem. According to the theorem, the rational roots of a polynomial equation can be found by considering all the possible factors of the constant term and dividing them by the factors of the leading coefficient. In this case, the constant term is -50 and the leading coefficient is 3. The factors of -50 are 1, 2, 5, 10, 25, and 50. The factors of 3 are 1 and 3. Therefore, the possible rational roots are: ±1/1, ±2/1, ±5/1, ±10/1, ±25/1, and ±50/1. Since the problem specifies that the rational roots should be within the range -2 ≤ x ≤ 5, we can eliminate the values outside this range. After finding the rational roots within the given range, we can substitute each value for x in the equation to check if they satisfy the equation.