Final answer:
The polynomial function f(x)=(x-4)(3x-1)² has two real roots, with one of them repeated due to the square, making it have two real and identical roots which are (x=4) and (x=⅓), corresponding to option (b).
Step-by-step explanation:
The polynomial function f(x)=(x−4)(3x−1)² can be analyzed to find its real roots by looking at its factors. The factor (x-4) gives us a root at x=4, which is a real root. The other factor (3x-1)² indicates that it is a square of a linear factor, giving us a real root at x=⅓. However, since this term is squared, it implies that the root at x=⅓ is repeated. Therefore, we have two distinct real roots, but one of the real roots has multiplicity two, meaning it counts as two roots at the same point on the x-axis. This matches option (b) that f has two real and identical roots.
The polynomial function f(x)=(x−4)(3x−1)² is a quadratic function since it can be written in the form ax²+bx+c = 0. To determine the nature of the roots of f(x), we need to analyze the discriminant of the quadratic equation.The discriminant, denoted by Δ, is given by the formula Δ = b²-4ac. If Δ > 0, the quadratic equation has two distinct real roots. If Δ = 0, the quadratic equation has two real and identical roots. If Δ < 0, the quadratic equation has two complex conjugate roots.For the given function f(x)=(x−4)(3x−1)², we can expand it to f(x) = 9x³ - 27x² + 25x - 4. By comparing this with the quadratic equation form, we can see that a = 9, b = -27, and c = -4. Plugging these values into the discriminant formula, we have Δ = (-27)² - 4 * 9 * (-4) = -27. Since Δ < 0, the function f(x) has one real root and one complex conjugate pair, which means the correct option is c) f has one real root and one complex conjugate pair.