Final answer:
The maximum number of non-real zeros possible for the function f(x) = 2.5^4−3x ^3 −2.6x ^2 - 5.1x−5.6 is four.
Step-by-step explanation:
The maximum number of non-real zeros possible for the function f(x) = 2.5^4−3x ^3 −2.6x ^2 - 5.1x−5.6 is four. The equation is a polynomial of degree four, which means it could have up to four complex solutions. Complex solutions occur when the discriminant of the quadratic equation is negative.
These roots can be real or complex numbers. Complex numbers, in this context, include non-real roots. Second, the Fundamental Theorem of Algebra states that every non-zero, single-variable, degree \( n \) polynomial with complex coefficients has, counted with multiplicity, exactly \( n \) complex roots.
Since the coefficients of our polynomial are real numbers, this theorem applies here as well. Third, for polynomials with real coefficients, like our function, complex roots must occur in conjugate pairs. A complex conjugate of a root is a root with the same real part and the opposite imaginary part.
Therefore, if a polynomial has a complex root \( a + bi \), it must also have the conjugate root \( a - bi \) as long as the coefficients of the polynomial are real numbers. Our function \( f(x) = 2.5x^4−3x^3−2.6x^2−5.1x−5.6 \) is a fourth-degree polynomial (since the highest power of \( x \) is 4). This means that \( f(x) \) has exactly 4 roots.
To determine the discriminant, we can use the formula b^2 - 4ac. In this case, a = -3, b = -2.6, and c = -5.1. Plugging these values into the formula, we get (-2.6)^2 - 4(-3)(-5.1) = 6.76 - 61.2 = -54.44. Since the discriminant is negative, there are four non-real zeros possible for the function.