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Polynomials are expressions that consist of variables raised to different powers, combined using addition, subtraction, and multiplication.
The roots of a polynomial are the values of the variable that make the polynomial equal to zero.
For instance, in the polynomial function, if you find a value of the variable, say \(x\), that makes \(f(x) = 0\), then \(x\) is a root of the polynomial.
Finding roots is essential because it helps in factoring the polynomial and simplifying expressions. In our example exercise, we are asked to determine if a complex number \(k = 2 - i\) is a root of the polynomial \(f(x) = 2x^3 - x^2 + 3x - 5\).

One efficient way to check if a given number is a root is using synthetic division, which checks the remainder when the polynomial is divided by \(x - (2 - i)\). If the remainder is zero, then that number is indeed a root.
This method is much faster and more straightforward than traditional polynomial division.

A complex number comprises a real part and an imaginary part, usually denoted as \(a + bi\), where \(i\) is the square root of -1.
When dealing with polynomials, complex numbers often come into play, especially when the polynomial has no real roots.
In the exercise, we're using \(k = 2 - i\), a complex number, to check if it is a zero of the polynomial. This means substituting \(2 - i\) in place of the variable and evaluating if \(f(2 - i) = 0\).

Synthetic division simplifies this process by breaking it down into smaller, more manageable steps.
You only need to perform basic arithmetic operations using the real and imaginary components separately.
Remember that imaginary units follow the rule \(i^2 = -1\), and this helps in simplifying the calculations.

Evaluating polynomial functions means calculating their value for a given input.
This is essential in many scenarios, like determining if a specific value is a root or analyzing the behavior of the polynomial.
In our example exercise, we use synthetic division to evaluate \(f(2 - i)\). If we find a remainder of zero, it confirms that \(2 - i\) is indeed a root.

Here's how synthetic division works:

• Write down the coefficients of the polynomial in descending order of power.
• Use the given complex number \(2 - i\) for synthetic division.
• Perform arithmetic operations to check if the remainder is zero.

Through synthetic division, you're essentially evaluating the polynomial in a step-by-step manner, ensuring all calculations are correct.
This method is powerful and quick, making it ideal for both real and complex numbers.
Ultimately, evaluating the polynomial helps in understanding its roots and overall structure, providing valuable insights.