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Polynomial long division is a process similar to long division with numbers, but instead, it's used for dividing polynomials. To break down complex polynomial expressions, long division can simplify the understanding of a polynomial's behavior, especially when finding factors or simplifying expressions.
The process involves dividing the polynomial's leading term by the leading term of the divisor and multiplying the entire divisor by that quotient. This product is then subtracted from the polynomial, and the process is repeated for the remaining polynomial terms. The end result consists of a quotient and possibly a remainder. If a polynomial divides another polynomial with zero remainder, then the former is a factor of the latter. An important use case, as shown in the exercise, is to determine specific values that can make a polynomial divisible by another.

Factoring polynomials is a critical concept in algebra that involves breaking down a polynomial into a product of its factors. These factors are simpler polynomials whose multiplication results in the original polynomial.
Strategies for factoring polynomials include finding the greatest common factor, using the difference of squares rule, sum and difference of cubes, or quadratic expressions. For more complex polynomials, factoring by grouping or applying special formulas might be needed. Factoring is not only a crucial step in solving polynomial equations but also in simplifying expressions and understanding the behavior of graphs. For the given exercise, understanding this concept helps us utilize the Remainder Theorem effectively, since a zero remainder after division indicates that the divisor is indeed a factor of the polynomial.

Synthetic division is an alternative to polynomial long division and is particularly useful when dividing by a linear factor. It's a shorthand method that allows for a quicker and more efficient calculation.
To perform synthetic division, you create a table using the coefficients of the polynomial to be divided, excluding any zero coefficients for terms that are missing. Then you bring down the leading coefficient and proceed to multiply and add according to the synthetic division algorithm, which closely mirrors the steps taken in long division. The result of synthetic division is the quotient and remainder, similar to long division. This method is especially convenient when applying the Remainder Theorem, as seen in our exercise, for it swiftly provides the remainder of the polynomial when divided by a linear factor, like the divisor used in our problem.