Final answer:
To find the real zeros of the function f(x) = 2.5x⁴ + 3x³ - 2.6x² - 5.1x - 5.6, we need to use the Rational Root Theorem to identify possible rational roots and test them to find the real zeros.
Step-by-step explanation:
To find the real zeros of the function f(x) = 2.5x⁴ + 3x³ - 2.6x² - 5.1x - 5.6, we need to solve the equation f(x) = 0. One way to do this is by using synthetic division or long division, but a more efficient method is to use the Rational Root Theorem to identify possible rational roots. In this case, the possible rational roots are factors of the constant term (±1, ±2, ±4, ±5, ±7, ±8, ±28). By testing these values, we can find the real zeros of the function.
We can start by testing x = 1 as a possible root. Plugging it into the function, we get f(1) = 2.5(1)⁴ + 3(1)³ - 2.6(1)² - 5.1(1) - 5.6 = 2.5 + 3 - 2.6 - 5.1 - 5.6 = -7.8. Since f(1) is not equal to 0, x = 1 is not a zero of the function.
We can continue testing the remaining possible rational roots until we find the real zeros of the function.