Problem 15 - College Algebra

Question

Factor by grouping. $$ 3 x^{3}-2 x^{2}-6 x+4 $$

Step-by-Step Solution

Answer

The factored form of the expression \(3 x^{3}-2 x^{2}-6 x+4\) by using the method of grouping is \((3x-2)(x^{2}-2)\).

Step 1: Rewrite the expression

Rewrite the expression in two groups. This is done by simply putting parentheses around the first and last two terms respectively like this: \((3 x^{3}-2 x^{2})-(6 x-4)\). We have added a minus sign in between to accommodate the minus sign before the third term in the original equation.

Step 2: Factor out the common terms

What we have to do next is to factorize each group separately. The first group, \(3 x^{3}-2 x^{2}\), has a common factor of \(x^{2}\), so we can take it out to get \(x^{2}(3x-2)\). The second group, \(-6 x+4\), has a common factor of 2, but we will factor out a '-2' because we expect matching terms in brackets that can be grouped further, so we have \(-2(3x-2)\). The expression now becomes \(x^{2}(3x-2) - 2(3x-2)\).

Step 3: Group the common factors

In this step, we find the same binomial factor in both terms, which is \((3x-2)\). This is our hint that we factored and grouped this correctly. Now factor out the shared binomial to get \((3x-2)(x^{2}-2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that involves breaking down a polynomial into simpler components or 'factors' that, when multiplied together, give back the original polynomial. It is akin to finding what ingredients were combined to bake a cake. A polynomial is an expression consisting of variables raised to non-negative integer powers and coefficients.

When factoring polynomials, we look for patterns or common factors in the terms. Polynomials can often be factored using various methods, such as taking out a greatest common factor, grouping terms (as in the exercise), using the difference of squares, or applying the quadratic formula when dealing with second-degree polynomials. Factoring is not only a pivotal concept in algebra but also a tool that simplifies expressions, making them easier to work with in equations and functions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and arithmetic operations. They can range from simple combinations, like \(2x + 3\), to more complex ones involving exponents, like the polynomial in our exercise. Understanding how to manipulate these expressions using algebraic rules is essential.

One of the manipulation techniques is factoring, which helps in simplifying expressions and solving equations. The goal is to transform complicated expressions into products of simpler expressions, which can reveal solutions to equations or insights into mathematical relationships.

Common Factors

A common factor is a number or algebraic term that divides exactly into two or more other numbers or terms. For example, in the exercise, \(x^{2}\) is a common factor in the terms \(3x^{3}\) and \(2x^{2}\). Similarly, ‘-2’ is a common factor in the terms \(-6x\) and \(4\), and it's deliberately used (despite being '-2') to reveal a matching binomial factor in both grouped terms.

Finding common factors is a foundational step in many factoring methods, allowing us to 'pull out' these shared components and simplify the expression. The skillful extraction of common factors lays the groundwork for simplification and the potential solution of algebraic equations.

Binomial Factors

Binomial factors are two-term expressions, such as \(3x - 2\) found in the exercise. In the process of factoring by grouping, we often look for such common binomial factors. A binomial factor is analogous to finding a pattern that repeats in different elements of the expression, and it serves as a key for further simplification.

When binomial factors are the same in multiple terms of an expression, as in our grouped expression \(x^{2}(3x-2) - 2(3x-2)\), it indicates that the factoring process is headed in the right direction. By recognizing and extracting these binomial factors, we can compress the expression into a product of simpler expressions, making it easier to evaluate or solve.