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A polynomial function is a type of function that involves only non-negative integer powers of the variable. For example, in the polynomial function given in the exercise, \(f(x)=x^{2}+3x+4\), you have terms with \(x^{2}\) and \(x\). The highest power of \(x\) in the polynomial is called the degree of the polynomial, which in our example is 2.
Polynomial functions can have various degrees:

• Linear: The degree is 1 (e.g., \(f(x) = 2x + 3\))
• Quadratic: The degree is 2 (e.g., \(f(x) = x^2 + 3x + 4\))
• Cubic: The degree is 3 (e.g., \(f(x) = x^3 + x^2 - x + 3\))

Polynomial functions are widely used in mathematics because they are simple and can be used to model many kinds of real-world phenomena. Knowing how to manipulate these functions, such as using synthetic division, is crucial for understanding their behavior and solving for their zeros.

The zeros of a polynomial are the values of \(x\) at which the polynomial evaluates to zero. In other words, they are the solutions to the equation \(f(x) = 0\). Identifying these zeros is a fundamental aspect of working with polynomial functions.
Let's delve into some important points about zeros:

• Real Zeros: These are real numbers where the polynomial equals zero.
• Complex Zeros: These are complex numbers (involving the imaginary unit \(i\)) where the polynomial also equals zero.

In the exercise, we are testing whether the complex number \(k = 2 + i\) is a zero of the polynomial \(f(x)=x^{2}+3x+4\). To do this, synthetic division is employed. If the remainder is zero, then \(k\) is a zero of the polynomial.
Finding the zeros of a polynomial helps in many applications, such as solving equations, graphing polynomial functions, and understanding their behavior.

Complex numbers extend the concept of real numbers by introducing the imaginary unit \(i\), which is defined by the property that \(i^2 = -1\). A complex number is usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
Here are key aspects of complex numbers:

• Real Part: The \(a\) in \(a + bi\)
• Imaginary Part: The \(bi\) in \(a + bi\)

In the given exercise, the complex number \(k = 2 + i\) is tested to determine if it is a zero of the polynomial function. To handle this, we use synthetic division, a method that simplifies polynomial division by focusing only on the coefficients.
Understanding how to work with complex numbers is crucial since they are used in numerous fields such as engineering, physics, and computer science. They provide insight into the behavior of polynomials and enable the solution of equations that real numbers alone cannot solve.