Problem 61 - Introductory Algebra for College Students

Question

Factor completely. $$9 x^{2}-6 x-24$$

Step-by-Step Solution

Answer

The factored form of the given quadratic expression is \((3x + 6)(3x - 4)\)

Step 1: Find two numbers

Find two numbers that multiply to \(9(-24) = -216\) and add to -6. These numbers are -12 and 18.

Step 2: Split the middle term

Split the middle term (-6x) into -12x and 18x. The equation becomes \(9 x^{2}-12x+18x -24\)

Step 3: Grouping

Group the terms to factor by grouping: \(3x(3x-4) + 6(3x-4)\)

Step 4: Factoring

Factor out the common binomial term to get the final solution: \((3x + 6)(3x - 4)\)

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler "factor" polynomials that multiply together to produce the original polynomial. It can be thought of as reverse distributing or identifying simpler forms within a complex expression. This technique is crucial in algebra as it simplifies complex expressions, making them easier to work with and understand.

In the given exercise, we are working on factoring a quadratic polynomial, which is any polynomial of the form \(ax^2 + bx + c\). The goal is to express this polynomial in a factored form such as \((px + q)(rx + s)\). Here, the coefficients and constants \(p, q, r,\) and \(s\) are determined through the factorization process, aimed at making the polynomial more manageable.

The challenge lies in finding the right pairs of numbers or expressions that will break down or simplify a polynomial efficiently. Factorization not only reduces the complexity of mathematical work but also aids in understanding the underlying properties of polynomial graphs. With step-by-step factorization, we gain a deeper insight into how each component of the polynomial interacts and contributes to the whole.

Binomial Terms

Binomial terms refer to algebraic expressions containing two distinct terms that are typically separated by a plus (+) or minus (-) sign. A binomial can be something like \(x + y\) or \(a - b\). In the context of our exercise, these terms are used when grouping or factoring the expression.

Binomial factorization often involves identifying common factors in grouped terms and rewriting a polynomial as a product of two binomials. In the exercise, after splitting the middle term of the original quadratic, we grouped terms like \(3x(3x-4) + 6(3x-4)\). This woodenly paves the way for the singular expression \((3x+6)(3x-4)\), where each set entails a binomial structure.

The significance of binomial terms in factorization isn't just about rewriting equations; it's about recognizing patterns within polynomials. Understanding binomial terms and how they combine can make complex polynomials easier to manage and solve. Getting familiar with operations like the distributive property transforms exercises into opportunities for deeper insights into algebraic principles.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) put together to represent values. Expressions can be as simple as a single variable like \(x\) or as complex as multiple terms combined with various operations \((x^2 - 6x - 24)\).

In factorization, understanding algebraic expressions means knowing how to manipulate and simplify them through valid algebraic operations. The exercise shows how algebraic expressions can be transformed from a complex quadratic polynomial to a product of binomials. This transformation requires recognizing the expression’s structure and applying appropriate techniques such as finding two numbers that multiply or adding specific terms.

• Understanding the structure: Recognizing the components of an expression (terms, coefficients, constants).
• Performing operations: Knowing how to add, subtract, and multiply terms appropriately.
• Rewriting expressions: Using techniques like factorization to express equations in different but equivalent forms.

Algebraic expressions form the backbone of most algebraic operations you will perform, so comprehending these building blocks is fundamental to mastering more complicated mathematical tasks.