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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols often represent numbers and quantities in formulas and equations. Understanding algebra is crucial because it forms the foundation for more advanced math topics.

One of the common tasks in algebra is solving polynomial equations. Polynomials are expressions consisting of terms, such as constants, variables, and exponents. For example, in the exercise, we have polynomials like \(2x^3 - 17x^2 + 54x - 68\) and \(x^2 - 6x + 9\). Using algebra, we can simplify these expressions and find solutions to polynomial equations through various methods, including polynomial long division.

Polynomial division is a method used to divide one polynomial by another. It is similar to the long division of numbers but involves variables and exponents. In the given exercise, we are dividing \(2x^3 - 17x^2 + 54x - 68\) by \(x^2 - 6x + 9\).

The goal is to simplify the division and find the quotient and remainder. Just like dividing numbers, polynomial division helps us simplify complex expressions and solve equations that would otherwise be tough to handle. This process involves multiple steps, where we divide, multiply, and subtract until we cannot divide any further.

The long division method is a step-by-step process to divide polynomials. Here is a simple guide to understand it better:

• Step 1: Set Up the Division
  Write the dividend (the polynomial to be divided) and the divisor (the polynomial you are dividing by) in the long division format. In our example, \(2x^3 - 17x^2 + 54x - 68\) is the dividend and \(x^2 - 6x + 9\) is the divisor.
• Step 2: Divide the Leading Terms
  Divide the first term of the dividend by the first term of the divisor. For our exercise, \( \frac{2x^3}{x^2} = 2x \).
• Step 3: Multiply and Subtract
  Multiply the result from Step 2 by the entire divisor and subtract it from the original polynomial to find the remainder. This step reduces the degree of the polynomial. Repeat this process with the new polynomial until you can no longer divide.
• Step 4: State the Quotient and Remainder
  Once the division is complete, the polynomial quotient and the remainder will be your final result.

This method helps break down complex polynomial division into simpler and more manageable steps.

The remainder is the part of the polynomial that is left after dividing as much as possible using the long division method. In our example, after performing polynomial long division:
\[ )\frac{2x^3 - 17x^2 + 54x - 68}{x^2 - 6x + 9} = 2x - 5 + \frac{6x - 23}{x^2 - 6x + 9} \]
The quotient is \ 2x - 5 \ and the remainder is \ 6x - 23 \.

Including the remainder is important to provide a complete solution to the polynomial division. If the remainder is zero, the divisor is a factor of the dividend. If not, the remainder must be written as a fraction over the original divisor, as shown in our exercise. This additional term shows how much is left after extracting the polynomial quotient.