Problem 59 Factor by grouping. $$ 6 x^{... [FREE SOLUTION] (2024)

Short Answer

Greatest Common Factor

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.