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Synthetic division is a simplified way to divide a polynomial by a binomial of the form \((x - k)\). It's much quicker and more straightforward than traditional long division, especially when performed with polynomials. We set up synthetic division by listing the coefficients of the polynomial and then follow a series of multiplication and addition steps to find the remainder. Here’s a simple way to understand it:

• Write down the coefficients of the polynomial in a row.
• Bring down the leading coefficient to start our calculations.
• Multiply it by \((k)\) and add this product to the next coefficient.
• Continue these steps until you’ve processed all coefficients.

The final value you get after these steps is the remainder, which, according to the remainder theorem, is the value of the polynomial at \((k)\). This method simplifies the otherwise complex task of polynomial division, making it a handy tool for students.

Polynomial functions are expressions involving a sum of powers in one or more variables multiplied by coefficients. For a single variable polynomial, it will look like: \[f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where each \(a_i\) is a coefficient, and \(n\) is a non-negative integer. They can be of any degree, from a simple constant to a complex expression involving high powers of \(x\). In our exercise, the polynomial is \(f(x) = x^2 - x + 3\):

• The coefficients are [1, -1, 3].
• These represent the terms \(x^2\), \(-x\), and the constant 3, respectively.

Polynomials are fundamental in calculus, algebra, and many areas of mathematics because they can approximate more complicated functions. Understanding how to manipulate and evaluate polynomials is a key skill in mathematics.

Complex numbers expand the concept of numbers beyond the real numbers to include \(i\), an imaginary unit where \(i^2 = -1\). A complex number is typically written in the form \((a + bi)\) where \(a\) and \(b\) are real numbers. In our exercise, we dealt with \(3 - 2i\):

• The real part is 3.
• The imaginary part is -2i.

When using complex numbers in problems like polynomial evaluations, we still follow the usual algebraic rules but with extra steps to manage the imaginary unit \(i\). During synthetic division with complex numbers, operations like multiplication and addition involve both the real and imaginary parts. Though they can look intimidating, they follow regular arithmetic rules, just with more terms to combine.