Question

Factor by grouping. $$ 6 x^{2}+21 x+8 x+28 $$

Short answer

The factored form is (2x + 7)(3x + 4).

Step-by-step solution

- Group the Terms

First, group the terms into two pairs: (6x^2 + 21x) + (8x + 28)

- Factor out the Greatest Common Factor (GCF) in Each Group

In the first group, the GCF is 3x. In the second group, the GCF is 4: 3x(2x + 7) + 4(2x + 7)

- Factor Out the Common Binomial

Notice that (2x + 7) is a common binomial factor in both groups. Factor it out: (2x + 7)(3x + 4)

Key concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in factoring algebraic expressions. It refers to the highest number or term that divides all terms in an expression without any remainder. For example, in the expression \(6x^2 + 21x\), the GCF is 3x. This is because 3x divides both 6x^2 and 21x without any remainder. Similarly, for the expression \(8x + 28\), the GCF is 4. Breaking down each term to its factors can help in identifying the GCF:

• 6x^2 = 2 × 3 × x × x
• 21x = 3 × 7 × x
• 8x = 2 × 2 × 2 × x
• 28 = 2 × 2 × 7

Look for the largest common factors within each expression. Once you identify the GCF, you can factor it out, simplifying the expression and making it easier to handle.

Binomial

A binomial is an algebraic expression that contains exactly two terms. Each term can be a number, a variable, or the product of numbers and variables. Examples of binomials include \(3x + 4\) and \(2x - 5\). In our exercise, the expression \(6x^2 + 21x + 8x + 28\) is first grouped into pairs, forming two binomials \(6x^2 + 21x\) and \(8x + 28\). This allows us to factor out the common binomial later.
Understanding binomials is crucial because many algebraic operations, including factoring by grouping, involve manipulating binomials. In our problem, we notice that \(2x + 7\) is a common binomial part in both factored groups \(3x(2x + 7) + 4(2x + 7)\), which helps simplify the final factorized form: \((2x + 7)(3x + 4)\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols (+, -, *, /). For instance, \(6x^2 + 21x + 8x + 28\) is an algebraic expression containing four terms. Understanding the components of algebraic expressions helps in performing various operations like factoring, simplifying, or expanding the expressions.
Identifying like terms within algebraic expressions can make manipulation easier. For example, the terms \(6x^2\), \(21x\), \(8x\), and \(28\) can be grouped to facilitate factoring. Recognizing components and performing correct steps is fundamental for solving algebra problems effectively.

Polynomials

A polynomial is a type of algebraic expression that includes terms in the form of \(ax^n\), where \(a\) is a coefficient, \(x\) is a variable, and \(n\) is a non-negative integer exponent. Polynomials can have one or multiple terms. For example, the exercise provided is a polynomial with four terms: \(6x^2 + 21x + 8x + 28\).
To factor polynomials by grouping, it helps to split the polynomial into smaller groups or binomials first, as seen in the given example: \((6x^2 + 21x) + (8x + 28)\). By finding the GCF and common binomial factors, the polynomial simplifies into two binomial factors: \((2x + 7)(3x + 4)\). Understanding polynomials and their structure is key to mastering various algebraic techniques, making problems easier to solve.