Problem 20 - Elementary Algebra

Question

Determine the GCF of all the terms. $$ 15 x(x+2), 9(x+2) $$

Step-by-Step Solution

Answer

The GCF is \(3(x+2)\).

Step 1: Write Down the Terms

The terms given are \(15x(x+2)\) and \(9(x+2)\). Our goal is to determine the Greatest Common Factor (GCF) of these expressions.

Step 2: Identify Common Factors in Coefficients

Break down the coefficients of each term. The coefficients are 15 and 9. The factors of 15 are 1, 3, 5, 15, and the factors of 9 are 1, 3, 9. The greatest common factor of 15 and 9 is 3.

Step 3: Identify Common Polynomial Factors

Look at the polynomial factors \((x+2)\). Both terms contain \((x+2)\), indicating that \((x+2)\) is a common polynomial factor.

Step 4: Combine Common Factors

The GCF of the numerical parts is 3 and the GCF of the polynomial part is \((x+2)\). Therefore, the GCF for the entire expressions is the product of these: \(3(x+2)\).

Key Concepts

Understanding Expressions

Expressions are mathematical phrases that can vary in complexity. They often consist of numbers, variables, and operations. In this context, terms like \(15x(x+2)\) and \(9(x+2)\) are expressions.
They are made up of components that describe mathematical relationships.

• They include numbers (like 15 and 9), variables (like \(x\)), and entire polynomial functions (like \((x+2)\)).
• Expressions do not have an equality sign, unlike equations, which makes them open-ended for manipulation.

Breaking them down into their components helps us simplify or solve problems involving these expressions.

Exploring Coefficients

Coefficients are the numerical part of terms in algebraic expressions. They tell you how much the term contributes to the expression.
In the examples \(15x(x+2)\) and \(9(x+2)\), the coefficients are 15 and 9, respectively.

• Coefficients are critical because they define the size of each term.
• They are important when determining common factors, as they need to be broken down to find the GCF.

Understanding coefficients allows us to easily spot and calculate shared factors between different terms.

Identifying Common Factors

Common factors in algebra are numbers or expressions shared by two or more terms. They simplify expressions by reducing terms based on their shared components.

• In the expression \(15x(x+2)\) and \(9(x+2)\), both terms include the polynomial factor \((x+2)\).
• They also share the coefficient factor of 3 when the numbers 15 and 9 are broken down.

Finding common factors is essential for simplification and for reducing expressions, enabling better problem solving.

Understanding Polynomial Factors

Polynomial factors involve expressions that include variables raised to powers or in groupings, like \((x+2)\). They form a major part of algebraic terms
Polynomial factors can drastically alter the nature of an expression due to their variable parts.

• For the terms \(15x(x+2)\) and \(9(x+2)\), \((x+2)\) serves as a common polynomial factor.
• These factors need to be identified and used alongside numerical factors (like 3) when calculating the GCF.

Understanding how polynomial factors work and combine gives vital insight into the mechanics behind algebraic simplification.