Problem 13 - Prealgebra

Question

Factor each expression. $$15+20 x$$

Step-by-Step Solution

Answer

The expression factors to 5(3 + 4x).

Step 1: Identify the Greatest Common Factor

First, we need to find the greatest common factor (GCF) of the coefficients in the expression. The coefficients are 15 and 20. The GCF of 15 and 20 is 5 since it is the largest number that divides both 15 and 20 without leaving a remainder.

Step 2: Factor Out the GCF

Next, we factor out the GCF from each term of the expression. We divide each coefficient by the GCF: - Divide 15 by 5 to get 3 - Divide 20 by 5 to get 4. So, factoring out 5 from the expression gives us: 5(3 + 4x).

Key Concepts

Greatest Common Factor (GCF)

A key concept in factoring is finding the Greatest Common Factor, often abbreviated as GCF. This is the largest number that can divide two or more numbers without leaving a remainder. Finding the GCF is a useful skill, especially when you're simplifying expressions or solving equations. Here’s how you can determine the GCF:

• List the factors of each number.
• Identify the common factors—those that are shared between the numbers.
• The greatest of these shared factors is the GCF.

For example, to find the GCF of 15 and 20: 15 is divisible by: 1, 3, 5, 15. 20 is divisible by: 1, 2, 4, 5, 10, 20. The common factor is 5, hence GCF(15, 20)=5. Once the GCF is identified, it can be factored out of the expression, simplifying it and making further operations easier.

Coefficients

Coefficients are the numerical part of the terms in an algebraic expression. They multiply the variables and play a crucial role in operations like factoring or simplifying. In the expression provided, 15 and 20 are the coefficients. When we talk about coefficients:

• They show how many times a term is counted.
• In algebraic expressions, they can be any real number, positive or negative, integer or fraction.

When factoring, always look at these coefficients to find and factor out the GCF. This simplifies the expression, as seen when you factor out the GCF from each coefficient to rewrite the expression in a simpler form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They are the building blocks of algebra and are used to express equations and inequalities.Observing the expression \(15 + 20x\):

• 15 is a constant term, which can stand alone without any variable.
• 20x is a term consisting of a coefficient (20) and a variable (x).

Combining these terms forms the complete expression. The main goal in working with algebraic expressions is often simplifying them, which frequently involves factoring.By factoring, you change the structure of the expression without altering its value. For "15 + 20x", factoring out the GCF leads to a simpler and more versatile expression: \(5(3 + 4x)\). This transforms the expression into a form that's often easier to work with for solving and understanding algebraic problems.