Problem 48 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ 33 h^{4}-22 $$

Step-by-Step Solution

Answer

The factored expression is \(11(3h^4 - 2)\).

Step 1: Understanding the Problem

We need to factor out the Greatest Common Factor (GCF) from the expression \(33h^4 - 22\). Factoring involves expressing the expression in a form of a product of factors.

Step 2: Identify the GCF

Find the GCF of the coefficients (33 and 22). The prime factors of 33 are 3 and 11, whereas the prime factors of 22 are 2 and 11. The common factor is 11. There are no variables in both terms, so the GCF is 11.

Step 3: Factor Out the GCF

Divide each term of the expression \(33h^4 - 22\) by the GCF which is 11. This means we compute \(\frac{33h^4}{11} - \frac{22}{11}\). This simplifies to \(3h^4 - 2\).

Step 4: Express the Factored Form

The factored expression is written as the product of the GCF and the simplified expression. Thus, the factored form of the original expression is \(11(3h^4 - 2)\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a mathematical concept used frequently in algebra to simplify expressions. The GCF of any two numbers is the largest number that can evenly divide both of them. When we talk about factoring out the GCF, we are essentially trying to simplify an expression into its most basic components.

Using the example from the original exercise, let's take the coefficients 33 and 22. To find the GCF, we need to discover the largest number that can divide both of these numbers without leaving a remainder. This means breaking down each coefficient into its prime factors.

• The prime factors of 33 are 3 and 11.
• The prime factors of 22 are 2 and 11.

By comparing the two, we see that the number 11 is common to both sets of prime factors, making it the GCF of 33 and 22.

Knowing how to find the GCF is essential because it helps reduce complex expressions into simpler ones, which is a key step in many algebraic problems.

Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. In the problem we are looking at, 33 is the coefficient of the term containing the variable, while 22 is the standalone number, acting as a constant term.

Understanding coefficients is important because they are directly involved when finding the GCF. In any mathematical expression that involves a variable (such as the term with the variable raised to a power), the coefficient tells you how many times that variable is being multiplied.

For instance, in the term \(33h^4\), 33 is the coefficient. It indicates that the expression includes 33 times the fourth power of \(h\). In contrast, the term -22 has no variable attached, making 22 its own coefficient.

When we factor expressions, we look to simplify by removing this numerical coefficient through division or common factor extraction. This makes working with the expression easier and is a fundamental technique in algebra that can be applied to simplify or solve more complex equations.

Prime Factors

Prime factors are the building blocks of a number. They are the prime numbers that multiply together to give the original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

Understanding prime factorization is vital for finding the GCF. Each number can be broken down into its prime factors, and the GCF is found by identifying the prime factors that appear in all numbers considered.

To illustrate, let's break down 33 and 22:

• For 33, the prime factors are 3 and 11.
• For 22, the prime factors are 2 and 11.

The prime factor that they have in common is 11, and therefore it is the GCF.

Grasping the concept of prime factors allows you to simplify subtraction or division steps when handling multiple terms, making it easier to manipulate and simplify equations. It's a core skill not just in algebra, but across all areas of mathematics.