Problem 15 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

Hence the factorized form by grouping of \(3x^3 - 2x^2 - 6x + 4\) is \((3x - 2) (x^2 - 2)\).

Step 1: Group the terms

The initial step to factoring by grouping is to divide the polynomial into two groups. Here, the given polynomial is \(3x^3 - 2x^2 - 6x + 4\). Accordingly, it can be divided into two groups: \(3x^3 - 2x^2\) and \(-6x + 4\).

Step 2: Factor out the Greatest Common Factor (GCF) from each group

For each group, factor out the Greatest Common Factor. For the first group \(3x^3 - 2x^2\), the common factor is \(x^2\). After factoring out \(x^2\), the expression becomes \(x^2 (3x - 2)\). Similarly, for the second group \(-6x + 4\), the common factor is \(-2\), factoring out which leads us to \(-2 (3x - 2)\).

Step 3: Factor out common binomial

The two terms of the polynomial now are \(x^2 (3x - 2)\) and \(-2 (3x - 2)\). Here, the common binomial factor is \((3x - 2)\). Factor out \((3x - 2)\) and it becomes: \((3x - 2) (x^2 - 2)\).

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is a crucial concept when it comes to polynomial factoring. Think of the GCF as the biggest number or expression that can evenly divide each term in a group, without leaving a remainder. This is similar to finding the greatest common divisor in basic arithmetic but applied to algebraic expressions.

The advantage of identifying the GCF is that it simplifies polynomials, making them more manageable.

• For example, take a look at the group of terms like \(3x^3\) and \(-2x^2\). By spotting the GCF, which is \(x^2\), you can factor it out to simplify the group to \(x^2(3x - 2)\).
• In another group, say \(-6x + 4\), \(-2\) is the greatest factor. By factoring \(-2\) out, you get \(-2(3x - 2)\).

Using the GCF helps in the vital first steps of factoring polynomials, making complex expressions easier to work with.

Polynomial Factoring

Polynomial factoring is like breaking down a complex object into simpler parts, a bit like dismantling a machine to understand how it functions. This technique is used to express a polynomial as a product of simpler polynomials or factors.

• In the exercise we have, the polynomial start is \(3x^3 - 2x^2 - 6x + 4\).
• By grouping terms and factoring the GCF from each group, we manage to transform the polynomial into an expression with a common binomial factor.

Grouping terms effectively reveals hidden structures within the polynomial. Step by step, once each group shares a common binomial, the expression becomes easier to handle. It turns into a multiplication of simpler polynomials, such as \((3x - 2)(x^2 - 2)\). Polynomial factoring can simplify problem-solving and even solve equations more efficiently.

Mastering this process involves practice and an understanding of how to identify patterns and common factors within given expressions.

Binomial Factor

A binomial factor refers to a polynomial with two terms, much like a simple mathematical phrase. Recognizing a common binomial factor within different parts of a polynomial is a powerful strategy in polynomial factoring.

Once you identify like terms in separate groups, the binomial factor becomes a shared term in the overarching polynomial.

• For example, you may notice \((3x - 2)\) repeatedly showing up after grouping and factoring the polynomial \(3x^3 - 2x^2 - 6x + 4\).
• In such cases, this commonality allows you to "factor it out", leading to a factorization such as \((3x - 2)(x^2 - 2)\).

Once this is done, not only is the polynomial factorized, but its roots (the values of \(x\) that make the polynomial equal to zero) become easier to find. Recognizing and using the binomial factor streamlines the factorization process and aids in unraveling complex polynomials.