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Chapter 0: Problem 1

$$\text { Factor out the greatest common factor.}$$ $$18 x+27$$

Short Answer

Expert verified
The factored expression of \(18x + 27\) by factoring out the greatest common factor is \(9(2x + 3)\).

Step by step solution

01

Determine the common factor of 18 and 27

First, identify all the factors for both numbers. The factors of 18 are 1, 2, 3, 6, 9, and 18, whereas the factors of 27 are 1, 3, 9, and 27. The common factors of 18 and 27 are 1, 3, and 9.
02

Identify the greatest common factor

Find the largest number that appears in both lists of factors. In this case, the greatest common factor between 18 and 27 is 9.
03

Factor out the greatest common factor

Now, factor out the greatest common factor (9) from the original expression by dividing both terms by the greatest common factor, which results in \(2x + 3\) inside the parentheses. Your factored expression becomes \(9(2x + 3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expression
An algebraic expression is a combination of constants, variables, and operators forming a mathematical phrase. Variables represent unknown values and can assume different numbers, while constants have a fixed numeric value. Operators, such as addition and subtraction, describe the mathematical operations to be performed on these numbers.

For example, in the expression \( 18x + 27 \right\backslash strong\)> the numbers 18 and 27 are constants, 'x' is a variable, and the '+' sign is an operator. These expressions are the fundamental building blocks of algebra and are used to describe relationships and to solve equations.
Greatest Common Factor (GCF)
The greatest common factor (GCF), also known as the greatest common divisor (GCD), of two or more numbers is the largest positive integer that divides all the numbers without leaving a remainder. Finding the GCF is a crucial step in various mathematical procedures, including simplifying fractions, and is particularly important when factoring algebraic expressions.

To find the GCF, as done in the given exercise, list the factors of each number and identify the largest factor that they have in common. In this case, for the numbers 18 and 27, the common factors were 1, 3, and 9, with 9 being the greatest. Understanding how to calculate the GCF helps simplify and solve more complex algebraic equations.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as the product of its factors, similar to breaking down a number into its prime factors. The process simplifies the expression and can aid in solving equations, graphing parabolas, and understanding the behavior of polynomial functions.

One method of factoring polynomials is to extract the greatest common factor from each term. For instance, the expression from our exercise, \( 18x + 27 \right\backslash strong\)> can be factored by recognizing that 9 is the largest factor of both terms. Consequently, factoring out the GCF leads to the simpler expression \( 9(2x + 3) \right\backslash strong\)>.
Basic Algebra
Basic algebra is the study of symbols and rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It involves operations with numbers and variables, following the order of operations, solving equations, factoring expressions, and working with functions.

For beginners, a solid grasp on basic operations such as addition, subtraction, multiplication, and division is essential, along with an understanding of how to manipulate expressions and solve for variables. The solution to factoring out the greatest common factor from an algebraic expression is an example of a foundational algebraic skill.

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Most popular questions from this chapter

$$\text { factor completely.}$$ $$-x^{2}-4 x+5$$

Find the intersection of the sets. $$\\{w, y, z\\} \cap \varnothing$$

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

Evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{2 x+3 y}{x+1}, \text { for } x=-2 \text { and } y=4$$

Fill in each box to make the statement true. Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. a. \(3^{\frac{1}{2}} 3^{\frac{1}{3}}\) b. \(\sqrt{7}+\sqrt{18} \quad \sqrt{7+18}\)
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