Problem 50 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 8(x+2)-y(x+2) $$

Step-by-Step Solution

Answer

The factored form is \((x+2)(8-y)\).

Step 1: Identify Common Factor

First, observe the polynomial expression: \(8(x+2) - y(x+2)\). Notice that \((x+2)\) is a common factor in both terms.

Step 2: Factor Out the Common Factor

Since \((x+2)\) is a common factor, factor it out from each term. The expression becomes \((x+2)(8 - y)\).

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is an essential concept in algebra that helps simplify expressions. It refers to the largest factor that divides each term in the polynomial without leaving a remainder. By identifying and factoring out the GCF, we simplify the expression, making it easier to work with. To find the GCF:

• Look at the coefficients (numbers in front of variables) and the variables themselves.
• Find the largest number that divides all coefficients.
• Identify any common variables in all terms, taking the lowest exponent present in each of them.

When we factor out the GCF, we essentially "distribute" it out, reducing the complexity of the polynomial.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding polynomials is crucial as they are foundational in algebra and calculus. A few key points about polynomials:

• They can have constants, variables, and exponents.
• The degree of a polynomial is the highest exponent of its variable.
• Polynomials can be classified based on their number of terms, like monomials (1 term), binomials (2 terms), and trinomials (3 terms).

In our exercise, both terms had the common binomial factor \((x+2)\), which we conveniently factored out. This step illustrates the utility of recognizing a polynomial's structure to simplify expressions.

Factoring Techniques

Factoring techniques help break down polynomials into products of simpler expressions. By using factoring, we can solve equations and simplify algebraic expressions more easily. Some basic factoring techniques include:

• Factoring out the GCF: Always the first step in simplification. Identify and extract the largest common factor.
• Factoring by grouping: A technique useful in polynomials with four or more terms. It involves grouping terms to find common factors.
• Difference of squares: Recognizes patterns like \(a^2 - b^2 = (a-b)(a+b)\).

In the original exercise, "factoring out the GCF," specifically the term \((x+2)\), simplified the expression to \((x+2)(8-y)\), demonstrating a straightforward application of this technique.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They represent values without specifying a particular value for the variables involved. Understanding how to manipulate and simplify algebraic expressions is vital to problem solving in algebra.Key features of algebraic expressions include:

• They can include constants and variables connected by operations such as addition, subtraction, multiplication, and division.
• The goal is often to simplify these expressions or solve them for particular variable values.

In the task at hand, the expression \(8(x+2) - y(x+2)\) included terms that could be simplified by recognizing a common factor. By factoring out \((x+2)\), we expressed the algebraic expression in a reduced form. Simplifying expressions like these is a fundamental skill in algebra that helps reveal insights about the relationships involving variables.