Answer :
Final answer:
The potential candidates for roots of the polynomial f(x) = 2x^4 + x^3 - 19x^2 + 9 are determined using the Rational Root Theorem. The possible rational roots are ±1, ±3, ±9, ±1/2, ±3/2, ±9/2.
Explanation:
To find potential candidates for roots of the polynomial f(x) = 2x4 + x3 - 19x2 + 9, we can use the Rational Root Theorem. According to this theorem, any rational root, expressed in its lowest terms p/q, has p as a factor of the constant term and q as a factor of the leading coefficient of the polynomial.
For the given polynomial, the constant term is 9 and the factors are ±9, ±3, and ±1. The leading coefficient is 2 and its factors are ±2 and ±1. Now, we form fractions by combining the factors of 9 with the factors of 2 to get the possible rational roots: ±1, ±3, ±9, and with negative signs, ±1/2, ±3/2, ±9/2.
To determine which of these are actual roots of the polynomial, one could use synthetic division or plug in the candidates to see if they satisfy the equation f(x) = 0.
