Problem 1 - College Algebra

Question

Factor out the greatest common factor. $$ 18 x+27 $$

Step-by-Step Solution

Answer

Factoring out the greatest common factor of the expression \( 18x + 27 \) gives \( 9 * (2x + 3) \).

Step 1: Identifying the greatest common factor

First, look at the coefficients of the variables, 18 and 27, and identify their greatest common factor. A common factor is a number by which both coefficients can be divided exactly. For 18 and 27, their greatest common factor is 9.

Step 2: Factoring out the greatest common factor

Having identified 9 as the greatest common factor of 18 and 27, the expression \( 18x + 27 \) can be re-written as \( 9 * 2x \) for the first term and \( 9 * 3 \) for the second term. This gives us \( 9 * 2x + 9 * 3 \)

Step 3: Simplifying the expression

The process is completed by 'factoring by grouping', where like-terms are factored together in a way that condenses the expression. Using basic rules of multiplication, the expression now becomes \( 9 * (2x + 3) \).

Key Concepts

Algebra

Algebra is a branch of mathematics that involves symbols and rules for manipulating those symbols to solve equations and understand relationships between quantities. It serves as the backbone for much of mathematics and its applications in modern science and technology. In algebra, expressions are a combination of numbers, variables, and arithmetic operations. Simplifying algebraic expressions, like the exercise given, involves using algebraic principles to rewrite expressions in a more concise form, making it easier for one to manage and solve related equations.

Factoring, a core algebraic technique, is a process of breaking down a complex expression into simpler factors that can be more easily interpreted or solved. In the given example, factoring the greatest common factor (GCF) is used to simplify the expression by identifying and dividing out the common factor from each term.

Factoring by Grouping

Factoring by grouping is an approach to factoring algebraic expressions that involves rearranging and grouping terms to find common factors. This method is particularly useful when dealing with polynomials that have four or more terms, but can also be applied to smaller expressions. The key step is to group terms in such a way that each group has a common factor. Once the groups are formed, the common factor is factored out, which often reveals a common binomial factor from the grouped terms.

In our simple exercise involving only two terms, the concept of grouping does not directly apply since we are only looking for the GCF, without the need to rearrange terms. However, the overarching idea is the same: we look for commonality that can be factored out to simplify the expression.

Simplifying Expressions

Simplifying expressions in algebra is all about making equations easier to understand and solve by reducing them to their simplest form. This usually means getting rid of parenthesis, combining like terms, and factoring, as appropriate. In the given exercise, we simplify the expression by factoring out the greatest common factor, which is identified in the first step of the provided solution. What this accomplishes is a reduction of complexity: the simplified expression, in this case, is in the form of a single term, a product of a number and a binomial.

Coefficients

In the world of algebra, coefficients are the numerical or constant factors that multiply the variables in an algebraic expression. They play a vital role in the manipulation and understanding of equations. For example, in the expression \(18x + 27\), '18' and '27' are the coefficients of the variable \(x\) and the constant term, respectively. Identifying the greatest common factor of these coefficients is the first step in simplifying the expression through factoring. This process requires finding the largest number that divides both coefficients without leaving a remainder. In our exercise, the number '9' is the GCF for '18' and '27', which we then factor out to streamline the original expression into a product involving a simpler binomial.