Problem 6 - Beginning and Intermediate Algebra

Question

Find the greatest common factor of each group of terms. $$24 r^{3} s^{6}, 56 r^{2} s^{5}$$

Step-by-Step Solution

Answer

The greatest common factor of terms \(24 r^{3} s^{6}\) and \(56 r^{2} s^{5}\) is \(8 r^2 s^5\).

Step 1: Find factors of the numerical part

To find the GCF of the numerical part, we will find the common factors of the coefficients (24 and 56). Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56 Let us identify the common factors among them:1, 2, 4, 8 The greatest common factor of 24 and 56 is 8.

Step 2: Find factors of the algebraic part

Now, from the algebraic part, we have r^3, r^2, s^6, and s^5. To find the GCF, we should select the minimum exponents of common variables. Let's compare exponents of r and s in both terms: - For variable r: r^3 and r^2 → the minimum exponent is r^2 - For variable s: s^6 and s^5 → the minimum exponent is s^5

Step 3: Combine the factors

Now, we combine the factors obtained in step 1 and step 2 to find the GCF: GCF = Numerical GCF × Algebraic GCF = 8 × r^2 × s^5 The greatest common factor of terms \(24 r^{3} s^{6}\) and \(56 r^{2} s^{5}\) is \(8 r^2 s^5\).

Key Concepts

Algebraic Expressions

An algebraic expression combines numbers, variables, and operations such as addition, subtraction, multiplication, and division.
These expressions represent quantities in a symbolic way and are a cornerstone in algebra.For example, in the problem we worked on, the expressions are:

• \( 24r^{3}s^{6} \)
• \( 56r^{2}s^{5} \)

Both expressions are great examples of algebraic expressions because they include numerical coefficients, variables \( r \) and \( s \), and they utilize the operation of multiplication.Understanding these elements helps in simplifying, factoring, and finding common factors among various terms in algebra.

Numerical Factors

Numerical factors are the numbers that can exactly divide a given number without leaving a remainder.
They are crucial when identifying the greatest common factor (GCF) of algebraic expressions, as seen with the coefficients like 24 and 56 in our example. Here's a quick breakdown:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56

From the list above, the largest number common to both sets is 8.
This indicates that 8 is the GCF for the numerical part of our given expressions.

Variable Exponents

Variable exponents in algebraic expressions indicate how many times a variable is multiplied by itself.
Understanding these exponents is key when finding the greatest common factor in expressions containing variables.Consider the variables from our terms:

• \( r^{3} \) and \( r^{2} \)
• \( s^{6} \) and \( s^{5} \)

To find the greatest common factor, we select the smallest exponent shared by each variable across the terms.
The minimum exponent of \( r \) is 2, so \( r^{2} \) is chosen.
For \( s \), the minimum exponent is 5, leading us to select \( s^{5} \).
This process ensures we are preserving the largest count of multiplied variables common to all expressions.

Factoring

Factoring in algebra involves breaking down an expression into simpler components, or factors, that when multiplied together, return the original expression.
This skill is particularly useful for simplifying expressions and solving equations.In the problem, we used the concept of factoring to find the greatest common factor (GCF) for both the numerical and algebraic parts:
1. Identified the GCF of the numerical coefficients (8 in this case).
2. Found the lowest common exponents for variables \( r \) and \( s \).We combine these factors:- The numerical part: 8- Variable part: \( r^{2}s^{5} \)Thus, the GCF of the expressions \( 24 r^{3} s^{6} \) and \( 56 r^{2} s^{5} \) is \( 8 r^{2} s^{5} \).Factoring helps simplify complex expressions and solve problems more efficiently.