Problem 61 - Algebra for College Students

Question

Factor by grouping. $$x^{2}+9 x+6 x+54$$

Step-by-Step Solution

Answer

The expression factors to \((x + 9)(x + 6)\).

Step 1: Group the Terms

The first step in factoring by grouping is to group the terms into two pairs. Here, you can group the terms as \((x^2 + 9x)\) and \((6x + 54)\).

Step 2: Factor Out the Greatest Common Factor

Now, factor out the greatest common factor (GCF) from each pair. For the first pair \((x^2 + 9x)\), the GCF is \(x\), which gives us \(x(x + 9)\). For the second pair \((6x + 54)\), the GCF is \(6\), which gives us \(6(x + 9)\).

Step 3: Factor Out the Common Binomial

Both terms now include the common binomial factor \((x + 9)\). Factor \((x + 9)\) out from both terms: \(x(x + 9) + 6(x + 9) = (x + 9)(x + 6)\).

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, commonly abbreviated as GCF, is a key concept in algebra, especially when factoring expressions. It refers to the highest number or expression that can divide two or more numbers or terms without leaving a remainder. Finding the GCF is crucial when simplifying algebraic expressions, as it helps to identify common factors that can be factored out of terms.
To find the GCF of terms in an algebraic expression, you will:

• Identify the variables and constants in each term.
• Find the highest power of each variable that appears in every term.
• Determine the largest integer that divides all the coefficients.

In our exercise, we applied the GCF concept twice. First, for the expression \(x^2 + 9x\), where the GCF was \(x\), and second, for \(6x + 54\), where the GCF was 6. These gave us the factored forms \(x(x + 9)\) and \(6(x + 9)\) respectively.

Binomial Expression

A binomial expression is a specific type of algebraic expression, consisting of exactly two terms. These terms are usually connected by a plus \( + \) or minus \( - \) operator. Binomials are common in algebraic manipulations, particularly during the process of factoring and expanding expressions.
In our example, the expression \(x^2 + 9x + 6x + 54\) was broken down into groups leading to the identification of the binomial \((x + 9)\) as a common factor. This demonstrates how binomials are often seen within larger polynomials during the factoring process.
Understanding binomial expressions is essential for factors like \((a + b)\) or \((a - b)\), which frequently appear in algebraic equations and expressions. Use of binomials helps to streamline and simplify expressions for better clarity and easier computation.

Algebraic Expressions

Algebraic expressions are composed of variables, constants, and operators (e.g., \(+, -, \times, \div\)). They serve as the foundational elements of algebra, allowing us to describe relationships and solve equations.There are different types of algebraic expressions, such as:

• Monomial: A single term expression.
• Binomial: An expression with two terms.
• Polynomial: An expression with multiple terms.

In our initial problem, \(x^2 + 9x + 6x + 54\) is a polynomial, as it contains four terms. Knowing how to manipulate and simplify these expressions is a core algebra skill.Factoring by grouping, as shown in this exercise, illustrates reducing complex polynomials by rearranging and combining terms for easier handling. It's an effective strategy for simplifying and finding solutions for equations involving large algebraic expressions.