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Algebraic fractions are much like the fractions you're used to, but instead of numbers in the numerator and denominator, they have polynomials. For instance, consider the fraction \( \frac{2x^2-18x-12}{x^3-4x} \). The concept is that an algebraic fraction can often be simplified or even solved for specific values, much like numerical fractions.

When it comes to algebraic fractions, one common goal is to simplify the expression. To do that, one might need to factor polynomials, cancel out common factors, or even perform partial fraction decomposition, especially useful when integrating rational functions or solving equations. Decomposition helps transform complex fractions into simpler, smaller pieces that are more manageable to work with.

Factorization is a crucial step in working with algebraic expressions, especially when dealing with partial fraction decomposition. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied back together, give you the original polynomial. This is akin to finding what numbers multiply together to give another number, such as \( 6 = 2 \times 3 \).

For example, consider the denominator from the exercise \( x^3-4x \). It can be factorized into simpler polynomials \( x(x^2-4) \), and then even further into \( x(x-2)(x+2) \). The goal is to get to these factors because they provide a starting point for the partial fraction decomposition. Without factorization, finding the partial fractions would be significantly more challenging, if not impossible.

Rational expressions are fractions where both the numerator and the denominator are polynomials. The exercise presented \( \frac{2x^2-18x-12}{x^3-4x} \) is an example of a rational expression. They often appear in calculus and algebra and can be intimidating due to their complexity.

Working with rational expressions frequently involves simplifying them by canceling common factors, if any, between the numerator and the denominator. Additionally, analyzing their behavior by studying their undefined values (where the denominator equals zero) is essential. In our example, after factoring and canceling the common factor \( x+2 \) from both the numerator and the denominator, the expression simplifies, making it easier to tackle further operations like partial fraction decomposition.

Polynomial division might sound daunting, but it's similar to long division with numbers. In the context of the partial fraction decomposition, you don't always have to perform a complete division; instead, you're typically on the lookout for common factors and simplifying the polynomial expression before breaking it down into its partial fractions.

However, in cases where the numerator is of higher degree than the denominator, you'd need to divide the polynomials first before partial fraction decomposition can be done. Thankfully, in our exercise, we avoided long division by canceling out a common factor which simplified our work. But if simplification was not possible, dividing the polynomials would be a necessary step before proceeding to find the partial fractions.