Problem 25 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(8 m-8\)

Step-by-Step Solution

Answer

8(m - 1)

Step 1: Identify the Greatest Common Factor (GCF)

Determine the greatest common factor of the terms in the polynomial. For the polynomial \(8m - 8\), both terms share a common factor of 8.

Step 2: Factor Out the GCF

Divide each term by the GCF and factor it out. Here, divide \(8m\) and \(-8\) by 8, which gives \(8(m - 1)\).

Step 3: Rewrite the Expression

Write the polynomial as the product of the GCF and the resulting polynomial. Therefore, \(8m - 8 \) factors to \(8(m - 1)\).

Key Concepts

greatest common factor

The greatest common factor (GCF) is the highest number that divides exactly into two or more numbers. It is a useful concept in various types of algebraic operations, including factoring polynomials.
To find the GCF of polynomial terms, follow these steps:

• List the factors of each term in the polynomial.
• Identify the highest common factor from these lists.

For example, in the polynomial iac 8m - 8iac, both terms have a common factor of 8. Recognizing the GCF is crucial before attempting to factor the polynomial.

polynomials

A polynomial is an algebraic expression that consists of variables, coefficients, and the arithmetic operations of addition, subtraction, and multiplication. Polynomials have the following nature:

• Each term in a polynomial consists of a coefficient and at least one variable.
• These terms are combined using the operations of addition and subtraction.

For example, iac 8m - 8iac is a polynomial because it contains two terms, each with a constant factor (8) and a variable factor (m). Understanding polynomials and their terms is the foundation for factoring them effectively.

factoring techniques

Factoring techniques are methods used to decompose polynomials into simpler components, which can then be multiplied to form the original polynomial. One key technique is factoring out the greatest common factor (GCF).
The process involves the following:

• Find the GCF of the polynomial's terms.
• Divide each term by the GCF.
• Rewrite the polynomial as the product of the GCF and the resulting polynomial.

In the provided exercise, factoring out the GCF 8 from the polynomial iac 8m - 8iac yields iac 8(m - 1)iac . This makes the expression simpler and more manageable.

algebraic expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. They are used to represent mathematical relationships and are fundamental to algebra.
An algebraic expression can be a single term or multiple terms, and it may include:

• Variables like iac m
• Constants like 8
• Operators like + and -

In our example, the expression iac 8m - 8iac involves a variable term (8m) and a constant term (8), separated by a subtraction operator. Factoring these expressions requires a good understanding of their structure and the relationships between their terms.