If \(\mathrm{f}(\mathrm{x})=4 \mathrm{x}^{5}+5 \mathrm{x}^{4}-8 \mathrm{x}^{3}+5 \mathrm{x}^{2} 4 \mathrm{x}-34 \mathrm{i}\) and \(\mathrm{f}[(-1+\sqrt{3} \mathrm{i}) / 2]=\mathrm{a}+\mathrm{ib}\) then \(\mathrm{a}: \mathrm{b}=\) (a) \(1: 2\) (b) \(-2: 1\) (c) \(17: 1\) (d) \(-17: 1\)

Understanding the nature of a polynomial function is essential when dealing with complex mathematical problems. Simply put, a polynomial function is a mathematical expression consisting of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Think of it as building blocks where the pieces are these operations.

For instance, the polynomial function in our exercise, \f\( f(x) = 4x^5 + 5x^4 - 8x^3 + 5x^2 + 4x - 34i \f\), is a quintic polynomial because the highest exponent is 5. This function is unique as it includes a complex number as the constant term, indicated by the \f\( i \f\) (the imaginary unit). Learning how to work with polynomial functions involves understanding how to perform operations with these terms, which may become quite intricate, especially when dealing with complex numbers.

When we talk about complex roots of a polynomial, we refer to the solutions of the polynomial equation that are not real numbers, but rather complex numbers. A complex number is of the form \f\( a + bi \f\), where \f\( a \f\) is the real part, \f\( b \f\) is the imaginary part, and \f\( i \f\) is the imaginary unit defined by \f\( i^2 = -1 \f\).

In our exercise, solving the polynomial function for specific complex roots requires substituting complex numbers into the function and simplifying. Finding these roots is crucial because by the Fundamental Theorem of Algebra, a polynomial of degree \f\( n \f\) has exactly \f\( n \f\) roots in the complex number system, including multiplicities. These roots can provide significant insights into the behavior of the polynomial function.

Complex arithmetic refers to mathematical operations applied to complex numbers. These operations include addition, subtraction, multiplication, division, and exponentiation, similar to arithmetic with real numbers, but with rules that address the unique properties of complex numbers.

For example, when multiplying complex numbers, you use the distributive property and the fact that \f\( i^2 = -1 \f\). In the given exercise, complex arithmetic is necessary to evaluate the polynomial's value at a particular complex number. Simplifying the expression involves calculating the powers of the complex number and adding or subtracting the terms appropriately. Becoming comfortable with these procedures is pivotal to understanding and successfully solving complex number operations within polynomial equations.

The ratio of complex numbers is represented by taking the real and imaginary components of one complex number and comparing them to the components of another, often using a colon to denote the ratio. In practice, this is similar to comparing the ratio of two real numbers, but with complex numbers, you deal with two parts: the real and imaginary components.

In the context of our exercise, we obtain a complex number after evaluating the polynomial at a given point, and we identify the ratio of the real part to the imaginary part. It's essential to remember that the imaginary unit, \f\( i \f\), is a fundamental part of the complex number and cannot be ignored. Simplifying the ratio involves recognizing common factors and reducing them to find the simplest form, which often provides a clearer understanding of the relationship between the real and imaginary parts of a complex number.