Problem 90 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$12 x^{2} y-27 y-4 x^{2}+9$$

Step-by-Step Solution

Answer

The fully factored form of the polynomial \(12x^{2}y - 27y - 4x^{2} + 9\) is \((2x - 3)(2x + 3)(3y - 1)\).

Step 1: Grouping

The polynomial is \(12x^{2}y - 27y - 4x^{2} + 9\). We can rearrange this into two groups: \(12x^{2}y - 27y\) and \(-4x^{2} + 9\). So it becomes: \(12x^{2}y - 27y - 4x^{2} + 9 = (12x^{2}y - 27y) - (4x^{2} - 9)\).

Step 2: Factor Out Common Terms

Factor out the common factor of \(3y\) from the first group and \(1\) from the second group. The equation becomes: \(3y(4x^2 - 9) - 1(4x^2 - 9)\).

Step 3: Further factorising

Observing now, we have a common term of \((4x^2 - 9)\) in both parts of the equation. Factoring out the common term in the equation, we get: \((4x^2 - 9)(3y - 1)\). Now, the first part of this equation \(4x^2 - 9\) is a difference of squares and can be factored further into \((2x - 3)(2x + 3)\).

Step 4: Final Factorization

Substituting \(4x^2 - 9\) with \((2x - 3)(2x + 3)\), the polynomial becomes: \((2x - 3)(2x + 3)(3y - 1)\). This is the complete factorization of the given polynomial.

Key Concepts

Factor by Grouping

Factoring by grouping is a method used to factor certain polynomials that have four terms. It involves rearranging the terms into two pairs and then finding common factors within each pair. To apply this technique, one must carefully group the terms so that each group has a factor in common, preferably in a manner that allows for further simplification. For example, in the polynomial 12x^2y - 27y - 4x^2 + 9, we rearrange terms into two groups (12x^2y - 27y) and (-4x^2 + 9) before proceeding. This step is essential because it paves the way for factoring out common factors in the next stage.

Correct grouping is crucial and sometimes requires insight into the structure of the polynomial. The use of factor by grouping is particularly effective when the polynomial contains a common factor in each group, allowing for the common factor to be factored out, simplifying the polynomial to a product of more manageable expressions.

Common Factor

Identifying a common factor involves looking for a term or expression that divides evenly into each part of a polynomial. When we find such a factor, it can be factored out, reducing the polynomial to a simpler form. In the example given, we focus on the grouped polynomial 3y(4x^2 - 9) - 1(4x^2 - 9), where the common factor is (4x^2 - 9). By recognizing this common factor, we can subsequently pull it in front of a second set of parentheses and place the remaining terms inside, resulting in (4x^2 - 9)(3y - 1).

Spotting a common factor can sometimes be the difference between a problem that seems insoluble and one that can be solved with relative ease. While in some cases the common factor may be a simple coefficient, other situations will present a binomial or more complex expression as the common factor, demanding a keen eye and algebraic understanding to identify.

Difference of Squares

The difference of squares is a special pattern in algebra that describes an expression written as the difference between two perfect squares, in the form a^2 - b^2. It can be factored into (a - b)(a + b). This concept is a powerful tool in polynomial factorization and is used when we encounter terms that fit the difference of squares form. For instance, in our equation, 4x^2 - 9 is recognized as a difference of squares because 4x^2 is the square of 2x, and 9 is the square of 3. This allows us to factor 4x^2 - 9 into (2x - 3)(2x + 3).

One must remember that the difference of squares only applies to subtraction. When students learn to spot this pattern, they can quickly factor expressions that might otherwise appear daunting. Mastering this concept opens the door to simplifying more complex polynomials involving squares.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of its factors, which are simpler polynomials or numbers. The ultimate goal of factorization is to represent the polynomial in its simplest form, where no further factoring is possible. For the given exercise, the final factorization of the polynomial 12x^2y - 27y - 4x^2 + 9 is arrived at by employing all the aforementioned methods: (2x - 3)(2x + 3)(3y - 1).

Polynomial factorization is not only a critical skill for solving equations and simplifying expressions but it also plays a significant role in calculus, such as in finding roots or zeroes of polynomials and in the integration of rational functions. It's one of the fundamental tools in a mathematician's toolkit and a building block for understanding higher-level concepts.