Problem 80 - Algebra A Combined Function

Question

Write each algebraic expression described. Write an expression of the form ________(________+__________)whose product is 6x + 24.

Step-by-Step Solution

Answer

The expression is \(6(x + 4)\).

Step 1: Identify Common Factors

Identify the greatest common factor in the given expression \(6x + 24\). Both terms, \(6x\) and \(24\), are divisible by 6. Therefore, 6 is the common factor.

Step 2: Factor Out the Common Factor

Factor 6 out of each term in the expression \(6x + 24\). This means you'll write 6 outside a set of parentheses, \(6(x + 4)\). Inside the parentheses, you divide each term by 6: \(x = \frac{6x}{6}\) and \(4 = \frac{24}{6}\).

Step 3: Write the Expression in Form \(a(b + c)\)

Now the expression is written in the desired form, which is \(6(x + 4)\). Here, \(a = 6\), \(b = x\), and \(c = 4\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the highest number that can evenly divide two or more numbers. In algebra, finding the GCF involves looking at the coefficients of terms within an expression. For our expression, which is \(6x + 24\), we break down the coefficients into their prime factors.

• The prime factors of 6 are 2 and 3.
• The prime factors of 24 are 2, 2, 2, and 3.

Looking at the factors, the common prime numbers are 2 and 3, whose product is 6.
This makes 6 the greatest common factor of the expression. Identifying the GCF is the crucial first step in simplifying algebraic expressions and solving many algebra problems. It helps in reducing algebra expressions to their simplest form.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations. For example, the expression \(6x + 24\) includes:

• A variable, \(x\), which represents an unknown quantity.
• Coefficients, such as 6 in \(6x\), which tell you how many times the variable is being multiplied.
• A constant, 24, which is a standalone number without a variable.

Algebraic expressions can come in many forms and are a fundamental part of understanding mathematics. They can represent many real-world situations and can be manipulated to solve equations, find patterns, or describe mathematical relationships.
Understanding the components of an algebraic expression allows us to perform operations such as addition, subtraction, or factoring.

Factoring Out Common Factors

Factoring out common factors is a process of simplifying an algebraic expression by dividing all terms by the greatest common factor. It’s a tool that helps in making expressions less complex and easier to work with. Given our expression \(6x + 24\), here is how we factor:First, identify the greatest common factor, which we found to be 6. Then, divide each term of the expression by this number and write what's left inside parentheses.

• For \(6x\), divide by 6 to get \(x\).
• For 24, divide by 6 to get 4.

Put the divide results together: \(6(x + 4)\).
This means you have factored out the common factor, 6, simplifying the original expression to a more manageable form. Factoring out common factors is not only helpful in simplifying expressions but also crucial in solving equations and can make solving them significantly easier.