The polynomial \( 3 x^{2}-11 x-8 \) is prime. True False

To determine if the polynomial \( 3x^{2}-11x-8 \) is prime, we need to check if it can be factored into two non-trivial polynomials. If it cannot be factored over the integers, it is considered prime. Let's compute the discriminant (\(D\)) of the polynomial, which is given by the formula \(D = b^2 - 4ac\), for \(a = 3\), \(b = -11\), and \(c = -8\): \[ D = (-11)^2 - 4(3)(-8) = 121 + 96 = 217 \] Since the discriminant is positive and not a perfect square, the polynomial doesn't factor nicely over the integers. Therefore, the polynomial \(3x^{2}-11x-8\) is indeed prime. False, the polynomial is actually prime. However, check if it can be factored further over the real numbers or other fields!