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Polynomial division is a method for breaking down complex polynomial expressions into simpler components. Think of it as a process similar to long division with numbers, but instead we're working with expressions that include variables such as \(x\). In our exercise, we are dealing with a polynomial division of the form \( \frac{x^{3}-9 x^{2}+27 x-27}{x-3} \). This means we want to find out

• how many times the divisor \((x-3)\) fits into the dividend \((x^3-9x^2+27x-27)\)
• and what remains once the division is complete.

Each time we "fit" the divisor into a section of the polynomial, we produce part of the quotient. Eventually, the coefficient values of the quotient, along with any leftover, give us the solution. This is what synthetic division helps us achieve in a more practical manner.

The terms "quotient" and "remainder" refer to the results received from dividing one polynomial by another. The quotient is the main result, or how many times the divisor fits into the dividend in polynomial terms. The remainder is what's left over after all possible times have been accounted for.
For our problem, when synthetic division is used, the quotient is represented by a new polynomial constructed from the bottom row of the synthetic division process. In this exercise, the quotient obtained is \(x^2 - 6x + 9\).
On the other hand, the remainder in polynomial division is what's left after the full division is done. Here, the remainder is 0, which means \(x-3\) divides \(x^{3}-9x^{2}+27x-27\) exactly with no left over. Having a remainder of 0 signifies that \((x-3)\) is a factor of the polynomial.

Coefficients in polynomials are the numerical values in front of the terms. They are vital as they determine how each term contributes to the polynomial's value. For example, in the polynomial \(x^3 - 9x^2 + 27x - 27\), the coefficients are 1, -9, 27, and -27, respectively.
These coefficients become key players in synthetic division. First, they are noted horizontally in order from the highest term to the constant. Next, computations occur as these coefficients are manipulated by the divisor, step by step, to ultimately yield the coefficients of the quotient. In our case, the new coefficients 1, -6, and 9 build the quotient polynomial \(x^2 - 6x + 9\). Understanding the role of coefficients can turn synthetic division into a manageable tool for polynomial arithmetic, revealing both the quotient and remainder efficiently.