Problem 104 - Elementary Algebra

Question

Factor out the GCF first and then factor by grouping. $$ 6 x 3-3 x 2-42 x+21 $$

Step-by-Step Solution

Answer

The factored form is \(3(x^2 - 7)(2x - 1)\).

Step 1: Factor Out the GCF

Identify the greatest common factor (GCF) of all the terms in the polynomial. We have the terms: \(6x^3, -3x^2, -42x, 21\). The GCF of these terms is 3. Factor out the GCF:\[3(2x^3 - x^2 - 14x + 7)\]

Step 2: Apply Factor by Grouping

Divide the polynomial inside the parentheses into two groups:First group: \(2x^3 - x^2\)Second group: \(-14x + 7\)

Step 3: Factor Each Group

For the first group \(2x^3 - x^2\), factor out \(x^2\):\[x^2(2x - 1)\]For the second group \(-14x + 7\), factor out \(-7\):\[-7(2x - 1)\]

Step 4: Combine the Groups

After factoring each group, we have:\[x^2(2x - 1) - 7(2x - 1)\]Notice both groups contain the common binomial factor \((2x - 1)\), so we can factor by grouping:\[(x^2 - 7)(2x - 1)\]

Step 5: Final Factored Form

The complete factored form after extracting the GCF and factoring by grouping is:\[3(x^2 - 7)(2x - 1)\]

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is a crucial concept in simplifying polynomials and other algebraic expressions. It is the largest factor that divides two or more numbers or terms without leaving a remainder. To determine the GCF of a polynomial, follow these steps:

• Identify all the terms in the polynomial.
• Find the greatest number that evenly divides all coefficients.
• Check the common factors of the variables, taking the smallest power for each variable.
• The product of these two results is the GCF.

For example, in the polynomial given, the terms are \(6x^3, -3x^2, -42x, 21\). The GCF of these coefficients (6, 3, 42, 21) is 3. Additionally, since there is no variable common to all terms, only the numeric GCF 3 is used to factor the polynomial. This results in the expression: \[3(2x^3 - x^2 - 14x + 7)\] Finding the GCF is the first step in simplifying complex expressions and helps in breaking them down for further manipulation, such as factor by grouping.

Factor by Grouping

Factor by grouping is a technique used to factor polynomials, especially those with four terms. This method focuses on creating groups of terms that have a common factor. Here's how to apply this technique effectively:

• Divide the polynomial into smaller groups. These groups should typically consist of terms that can be factored together.
• Factor out the greatest common factor from each group.
• Look for a common binomial factor between the groups.
• Combine these groups using the common factor once more to reach a simplified expression.

In the polynomial \(2x^3 - x^2 - 14x + 7\), we can group terms like this:First group: \(2x^3 - x^2\) enables factoring out \(x^2\) resulting in \(x^2(2x - 1)\).Second group: \(-14x + 7\) allows factoring out \(-7\), leading to \(-7(2x - 1)\).Having factored each group, both now share a common factor of \((2x - 1)\). Using the common factor, we can write:\[(x^2 - 7)(2x - 1)\]This approach effectively simplifies the polynomial into an easily factorable product, a vital step in polynomial factorization.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like addition and multiplication). These expressions represent relationships between quantities and are foundational in algebra for solving equations and simplifying problems.

• Expressions vary in complexity from simple (e.g., \(x + 3\)) to complex (e.g., \(3x^2 + 5xy - 4\)).
• Understanding each part of an expression is key to effective problem-solving.
• Terms in algebraic expressions can take various forms, including constants, variables, coefficients, and powers or exponents.

For effective simplification, each component of the expression must be understood separately. Combining or factoring terms often requires identifying similar attributes or shared factors. In the given problem, understanding the composition of the terms \(6x^3, -3x^2, -42x, 21\) allows us to employ strategies like factoring and grouping. These help in breaking down and simplifying the complex expression into more manageable parts. Algebraic expressions serve as the building blocks for various mathematical operations critical for algebraic manipulations.

Polynomial Factorization

Polynomial factorization is a process of breaking down a polynomial into the product of its factors. This process is analogous to breaking a number into its multiplicative components. Understanding polynomial factorization is integral in simplifying expressions and solving polynomial equations. Here's a simple breakdown:

• Identify if the entire polynomial can be factored by a single term, such as a GCF.
• Look for specific techniques, such as factor by grouping, to simplify further.
• Once grouped and factored, check if the remaining factors can be factored further.
• If completed correctly, the result will be a product of polynomials or numbers, resulting in the expression's simplest form.

In this exercise, we factor the polynomial \(6x^3-3x^2-42x+21\) by first extracting the GCF, producing \(3(2x^3 - x^2 - 14x + 7)\). Next, factor by grouping breaks it into \((x^2(2x - 1) - 7(2x - 1))\), which further simplifies to \((x^2 - 7)(2x - 1)\). Thus, the fully factored form becomes \(3(x^2 - 7)(2x - 1)\).Polynomials can often be challenging, but mastering factorization techniques simplifies a vast array of algebraic problems, making them more approachable and solvable.