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Polynomial division is a method for dividing a polynomial by another polynomial. It is similar to long division, but it specifically applies to expressions involving variables raised to powers, such as \(ax^2 + bx + c\). The purpose of polynomial division is to simplify complex polynomial expressions and solve polynomial equations.
Unlike basic arithmetic, polynomial division deals with terms like \(x^2, x\), and constants:

• The dividend is the polynomial you are dividing.
• The divisor is the polynomial by which you are dividing.

The result can be expressed as the quotient plus the remainder. When using synthetic division, we typically deal with only linear binomials for simplicity. Synthetic division provides a quicker and simpler way to divide, especially when compared to the long division method.

A linear binomial is an expression of the form \(x - c\), where \(c\) is a constant. It contains exactly two terms, making it 'binomial', and the highest power of the variable \(x\) is one, making it 'linear'.
Examples of linear binomials include: \(x - 3, x + 5, x - \frac{1}{2}\).

• In synthetic division, the linear binomial must always be in the form \(x - c\).
• The value of \(c\) is critical as it is used throughout the synthetic division process.

To use synthetic division correctly, always ensure your divisor fits this format, as shown in the exercise.

Standard form refers to writing a polynomial with its terms in descending order of their degrees, from highest to lowest. For a polynomial to be in standard form, it should look like this: \[a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\]
This arrangement helps to maintain consistency and accuracy during division:

• \(a_n, a_{n-1}\), etc., are the coefficients of the polynomial terms.
  The term \(a_n x^n\) represents the highest degree of the polynomial.
• Making sure the polynomial is in standard form enables straightforward application of the division methods.

Always rewrite the polynomial in standard form before performing synthetic division to ensure the process works smoothly.