Problem 82 - Introductory Algebra for College Students

Question

Factor completely. $$24 x^{2}+3 x y-27 y^{2}$$

Step-by-Step Solution

Answer

The completely factorized expression is \(3(x - y)(8x + 9y)\)

Step 1: Identify common factors

The first thing to notice in the expression \(24x^{2} + 3xy - 27y^{2}\) is that all coefficients are divisible by 3. Thus, we can factor out 3 from the expression. This gives: \(3(8x^{2} + xy - 9y^{2})\).

Step 2: Perform trinomial factorization

Now observe the trinomial expression \(8x^{2} + xy - 9y^{2}\). We need to find two numbers that multiply to \(8*(-9) = -72\) (Product of a and c in the quadratic equation ax^2 + bx + c) and add up to 1 (Coefficient of the middle term). The two numbers meeting these conditions are 9 and -8.

Step 3: Split the middle term

Next, we have to rewrite the trinomial by splitting the 'xy' term into two terms using the two numbers found in step 2. This leads to: \(8x^{2}+ 9xy -8xy- 9y^{2}\)

Step 4: Group and factorize

Then we group the terms and factorize the binomials. It is noticeable that we can factor both the first two terms and the last two terms. This will lead to: \(x(8x + 9y) - y(8x + 9y)\)

Step 5: Further factorize any common factors

Afterwards, factorize any common binomial found in the expression which gives \( (x - y)(8x + 9y)\). Recall that at step 1 we took out a common factor of 3. So, the final answer is \(3(x - y)(8x + 9y)\)

Key Concepts

Trinomial Factorization

Trinomial factorization is an important process in algebra where we break down a trinomial into the product of two simpler binomials. A trinomial is a polynomial with three terms, such as \( ax^2 + bx + c \). The goal of trinomial factorization is to express this trinomial as the product of two binomials like \((mx+n)(px+q)\).
To factor a trinomial, we look for two numbers that multiply to give \(a \times c\) (the product of the leading coefficient and the constant term) and add up to \(b\) (the middle coefficient). This step aids in splitting the middle term to prepare it for grouping and factoring by parts.
In practice, with our example \(8x^{2} + xy - 9y^{2}\), we found numbers 9 and -8, which total 1, and allowed us to split the middle term into \(9xy - 8xy\), leading to effective grouping.

Quadratic Equations

Quadratic equations are essential in algebra, typically represented in the form \(ax^2 + bx + c = 0\). These equations can define parabolas and find real-world applications from physics to finance. While solving them can involve several methods such as the quadratic formula or completing the square, factorization remains a straightforward method when applicable.
Factorization is often favorable if the quadratic polynomial can be broken into two binomials easily (i.e., factors can be readily found). The equation \(8x^{2} + xy - 9y^{2} = 0\) illustrates using factorization, allowing us to find the roots of the equation or simplify expressions within a larger algebraic context.
Recognizing when a polynomial fits the quadratic format helps in understanding which approach to employ for factorization and further calculations.

Common Factors

Common factors are numbers or algebraic expressions that divide all terms of a given polynomial without a remainder. Identifying common factors is critical in simplifying polynomials, as it reduces complexities in equations and expressions.
In the polynomial \(24x^{2} + 3xy - 27y^{2}\), identifying common factors is the first step. Here, each term's coefficients are divisible by 3, allowing us to factor out 3, simplifying the expression to \(3(8x^{2} + xy - 9y^{2})\).
This reduction provides a cleaner and simpler polynomial that can be effectively dealt with in further factorization steps, enhancing understanding and reducing computational challenges. Always begin with this step when handling polynomials, as it maximizes simplicity and efficiency.