Problem 8 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ x^{3}, x^{2}, x^{5} $$

Step-by-Step Solution

Answer

The GCF is \(x^2\).

Step 1: Identify the Variables

The numbers to consider for finding the GCF are the variables in each term, specifically the powers of the variable 'x'. The terms given are \(x^3, x^2,\) and \(x^5\).

Step 2: Determine the Lowest Power

To find the GCF, we need the lowest power of 'x' that appears in all terms. The powers of 'x' in the given terms are 3, 2, and 5. The smallest of these powers is 2.

Step 3: Write the GCF

Based on the lowest power identified, the greatest common factor (GCF) of the terms \(x^3, x^2,\) and \(x^5\) is \(x^2\).

Key Concepts

Variables

In algebra, variables play a crucial role as they represent unknowns or quantities that can vary. In our scenario, the variable in the set of terms is "x". Understanding variables involves knowing:

• Variables are placeholders for numbers. They can change their value depending on the context.
• In the expression like \( x^3 \), "x" acts as a variable while "3" is its exponent.
• Variables can be tied to each mathematical operation and follow specific rules when manipulating expressions.

When dealing with variables in expressions, it is important to identify them correctly, as they are the key elements to consider while finding the GCF. Recognizing variables quickly will help you in simplifying and solving algebraic problems.

Exponents

Exponents are mathematical symbols used to represent the number of times a variable or a number is multiplied by itself. Understanding exponents is foundational for working with polynomial expressions:

• The exponent in an expression like \( x^3 \) is "3", indicating that "x" is multiplied by itself 3 times: \( x \times x \times x \).
• When finding the GCF among terms with the same base variable, you focus on the exponents, as they determine the power of each term.
• Exponents follow certain mathematical rules, such as \( x^a \times x^b = x^{a+b} \) and \( (x^a)^b = x^{a \times b} \).

In this exercise, identifying the smallest exponent among the terms is vital as it helps determine the GCF. The exponents in \( x^3, x^2, \) and \( x^5 \) are key to solving the exercise effectively.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and constants, arranged in terms of powers, and connected through addition or subtraction. Each part of a polynomial is called a 'term':

• An example of a polynomial expression is \( 2x^3 + 3x^2 - x + 5 \), where each part \( 2x^3, 3x^2, -x, \) and \( 5 \) are individual terms.
• In our exercise, the polynomial expressions are singular terms: \( x^3, x^2, \) and \( x^5 \) which illustrate the multiplication of the variable "x" and its exponents.
• The greatest common factor of polynomial terms is found by comparing variables and their lowest powers. This helps simplify expressions or solve equations easily.

Understanding polynomial expressions involves recognizing how terms are constructed and manipulated. Solving for the GCF requires assessing each term's components (both coefficient and variable parts). Recognizing these elements makes complex expressions manageable.