Applying The Distributive Property To Factor $12x^3 - 9x^2 + 4x - 3$

In the realm of algebra, the distributive property serves as a cornerstone for simplifying and manipulating expressions. It allows us to multiply a single term by multiple terms within a set of parentheses, effectively expanding or factoring expressions. When faced with the task of factoring a polynomial like $12x^3 - 9x^2 + 4x - 3$, the distributive property becomes an indispensable tool. This article delves into the process of applying the distributive property to factor this specific expression, evaluating different options, and identifying the correct approach.

Understanding the Distributive Property

Before we tackle the problem at hand, let's briefly revisit the distributive property. In its simplest form, the distributive property states that for any numbers a, b, and c:

$a(b + c) = ab + ac$

This principle extends to polynomials, where 'a' can be a monomial and '(b + c)' can be a polynomial with multiple terms. The key is to multiply 'a' by each term inside the parentheses.

Factoring, in essence, is the reverse of distribution. We aim to identify common factors within an expression and extract them, rewriting the expression as a product of factors. When dealing with polynomials like $12x^3 - 9x^2 + 4x - 3$, we often look for grouping opportunities where the distributive property can be applied in reverse.

Analyzing the Given Options

The question presents us with several options, each representing a different way to potentially rewrite the expression $12x^3 - 9x^2 + 4x - 3$ using the distributive property. Our goal is to determine which option facilitates factoring the expression effectively. Let's examine each option closely:

Option A: $3(4x^3 - 1) - (9x^2 + 4x)$

Option A presents the expression as $3(4x^3 - 1) - (9x^2 + 4x)$. Let's apply the distributive property to the first term: $3 * 4x^3 = 12x^3$ and $3 * -1 = -3$. So, the first part becomes $12x^3 - 3$. The expression then is $12x^3 - 3 - 9x^2 - 4x$. Rearranging the terms, we get $12x^3 - 9x^2 - 4x - 3$. This looks similar to our original expression, but it does not readily reveal a common factor across all terms. The terms are grouped in a way that doesn't immediately suggest a factorization pathway. Applying the distributive property here does expand a portion of the expression, but it doesn't lead us closer to a factored form.

Option B: $4x(3x^2 + 1) - 3(3x^2 - 1)$

Option B gives us $4x(3x^2 + 1) - 3(3x^2 - 1)$. Applying the distributive property to both parts, we get: $4x * 3x^2 = 12x^3$, $4x * 1 = 4x$, $-3 * 3x^2 = -9x^2$, and $-3 * -1 = +3$. Combining these, the expression becomes $12x^3 + 4x - 9x^2 + 3$. Rearranging the terms, we have $12x^3 - 9x^2 + 4x + 3$. Notice that the constant term is $+3$ instead of $-3$ as in our original expression. Therefore, this option does not represent a correct rearrangement of the original polynomial using the distributive property.

Option C: $3x(4x^2 - 3) + 1(4x - 3)$

Option C presents the expression as $3x^2(4x - 3) + 1(4x - 3)$. Let's apply the distributive property and see what we get. First, expand $3x(4x^2 - 3)$: $3x * 4x^2 = 12x^3$ and $3x * -3 = -9x^2$. The first part becomes $12x^3 - 9x^2$. The expression then transforms to $12x^3 - 9x^2 + 4x - 3$. Now, notice that we have a common binomial factor of $(4x-3)$ in both terms. This is a significant observation, as it directly aligns with the goal of factoring.

Option D: $3x^2(4x - 3) + 1(4x - 3)$

Option D presents the expression as $3x^2(4x - 3) + 1(4x - 3)$. Applying the distributive property in reverse, we can see that the term $(4x - 3)$ is common to both parts of the expression. We can factor out this common term: $(4x - 3)(3x^2 + 1)$. Expanding this factored form confirms that it is equivalent to the original expression: $3x^2 * 4x = 12x^3$, $3x^2 * -3 = -9x^2$, $1 * 4x = 4x$, and $1 * -3 = -3$. Thus, the expanded form is $12x^3 - 9x^2 + 4x - 3$, which matches our initial expression. This option successfully applies the distributive property to reveal a factored form of the polynomial.

Identifying the Correct Application

After analyzing each option, it becomes clear that Option D is the correct choice. Option D, $3x^2(4x - 3) + 1(4x - 3)$, successfully applies the distributive property to reveal a common binomial factor, allowing us to factor the expression as $(4x - 3)(3x^2 + 1)$. The other options either do not lead to a factorization or result in an expression different from the original.

Step-by-Step Factoring Process

Let's solidify our understanding by outlining the step-by-step process of factoring the expression using Option D:

1. Rewrite the expression using Option D: $12x^3 - 9x^2 + 4x - 3 = 3x^2(4x - 3) + 1(4x - 3)$

2. Identify the common binomial factor: The common binomial factor is $(4x - 3)$.

3. Factor out the common binomial factor: $(4x - 3)(3x^2 + 1)$

Therefore, the factored form of $12x^3 - 9x^2 + 4x - 3$ is $(4x - 3)(3x^2 + 1)$.

Conclusion

The distributive property is a powerful tool in algebra, enabling us to both expand and factor expressions. When factoring polynomials, it's often necessary to explore different groupings and applications of the distributive property to identify common factors. In the case of $12x^3 - 9x^2 + 4x - 3$, Option D, $3x^2(4x - 3) + 1(4x - 3)$, correctly demonstrates how the distributive property can be applied to reveal the factored form $(4x - 3)(3x^2 + 1)$. By carefully analyzing the options and applying the distributive property methodically, we can effectively factor complex algebraic expressions.

This process highlights the importance of recognizing patterns and strategically applying algebraic principles to simplify and manipulate expressions. Factoring, in particular, is a crucial skill in algebra, with applications in solving equations, simplifying rational expressions, and more.