Problem 17 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ -10 x^{2}, 15 x^{3} $$

Step-by-Step Solution

Answer

The GCF is \(5x^2\).

Step 1: Find Prime Factorization

First, let's find the prime factorization of each coefficient. The coefficient of \(-10 x^2\) is \(-10\), which factors into \(-1 imes 2 imes 5\). For \(+15 x^3\), the coefficient \(+15\) factors into \(3 imes 5\). Both coefficients have a \(5\) in common.

Step 2: Extract the Variable Part

Now, look at the variable parts \(x^2\) and \(x^3\). The common variable part is \(x^2\) because \(x^2 \) is the smallest power common to both terms.

Step 3: Combine the Common Factors

Combine the common factors found in the previous steps. The common factor of the coefficients is \(5\), and the common variable is \(x^2\). Therefore, the GCF is \(5x^2\).

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its basic components, which are prime numbers. Prime numbers are numbers greater than one that have no divisors other than 1 and themselves. For example, 2, 3, 5, and 7 are all prime numbers.
In the context of finding the Greatest Common Factor (GCF), prime factorization helps you identify the shared factors among numbers. For the example in the exercise,

• The number \(-10\) breaks down to \(-1 \, 2,\) and \(5\).
• The number \(15\) breaks down to \(3, 5\).

The common prime factor between the coefficients \(-10\) and \(15\) is \(5\). Finding these common factors using prime factorization helps simplify complex problems and is a crucial step in determining the GCF.

Algebraic Expressions

An algebraic expression consists of variables, numbers, and operations combined to represent a mathematical concept. In the given exercise,

• \(-10x^2\) is an algebraic expression with \(-10\) as a coefficient and \(x^2\) as the variable part.
• Similarly, \(15x^3\) has \(15\) as the coefficient and \(x^3\) as the variable part.

These expressions can be complex, having multiple terms or simple, having a single term. Breaking them into parts helps you focus and solve specific elements. Understanding how to factorize its parts in terms of coefficients and variable powers is key when working on problems. Algebraic expressions are fundamental in mathematics for conveying information or equations more compactly.

Variable Factors

Variable factors are the variable components in an algebraic term that can be represented in powers. In the exercise, you see expressions like \(x^2\) and \(x^3\). Each of these expressions has variable factors that are critical for determining the GCF.
When comparing two terms, focus on the variable portion:

• For \(x^2\) and \(x^3\), the smaller power, \(x^2\), is the common factor.

By understanding variable factors clearly, you can determine which variable components can be combined or simplified. Identifying the lowest power common to all the terms is essential to finding the GCF involving variables. Remember, the GCF will always take the smallest power of the shared variable factors across terms.