Factor 26r3s 52r5 39r2s4 What Is The Resulting Expression

Factoring Expressions: A Comprehensive Guide to Finding the Greatest Common Factor (GCF)

Factoring algebraic expressions is a fundamental skill in algebra, crucial for simplifying expressions, solving equations, and understanding more advanced mathematical concepts. This article delves into the process of factoring, focusing on how to find the greatest common factor (GCF) and apply it to factor expressions efficiently. We'll explore various techniques and provide a step-by-step approach to tackling complex expressions like the one presented: 26r³s, 52r⁵, and 39r²s⁴.

Understanding the Greatest Common Factor (GCF)

Before embarking on factoring, grasping the concept of the GCF is paramount. The GCF of a set of numbers or terms is the largest factor that divides each of them without leaving a remainder. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.

Finding the GCF involves several steps:

1. Prime Factorization: Break down each number into its prime factors. Prime factors are numbers divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).

2. Identify Common Factors: Compare the prime factorizations of the numbers. Identify the factors that appear in all the numbers.

3. Multiply Common Factors: Multiply the common factors together to find the GCF.

Let's illustrate this with an example: Find the GCF of 24, 36, and 48.

• 24: 2 x 2 x 2 x 3 = 2³ x 3
• 36: 2 x 2 x 3 x 3 = 2² x 3²
• 48: 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

The common factors are 2² and 3. Multiplying these gives us 2² x 3 = 4 x 3 = 12. Therefore, the GCF of 24, 36, and 48 is 12.

Factoring Algebraic Expressions with Variables

When factoring algebraic expressions containing variables, the process extends to include the variables. We look for the GCF of the numerical coefficients and the lowest power of each variable present in all terms.

Consider the expression 12x²y + 18xy² + 6xy.

1. Numerical Coefficients: The coefficients are 12, 18, and 6. The GCF of 12, 18, and 6 is 6.

2. Variables: The variables are x and y. The lowest power of x present in all terms is x¹, and the lowest power of y is y¹.

3. GCF of the Expression: Combining the numerical and variable GCFs, we get 6xy.

Therefore, the factored expression is 6xy(2x + 3y + 1).

Factoring the Expression: 26r³s, 52r⁵, 39r²s⁴

Now, let's apply these principles to the given expression: 26r³s, 52r⁵, and 39r²s⁴. Our goal is to find the GCF of these three terms.

1. Numerical Coefficients: The coefficients are 26, 52, and 39. Let's find their prime factorizations:

   • 26 = 2 x 13
   • 52 = 2 x 2 x 13 = 2² x 13
   • 39 = 3 x 13

   The common factor is 13.

2. Variable 'r': The powers of 'r' are 3, 5, and 2. The lowest power is r².

3. Variable 's': The powers of 's' are 1, 0 (since 's' is not present in 52r⁵), and 4. The lowest power is s⁰ which is equivalent to 1. Because 's' is not present in every term, it will not be a factor in the GCF.

4. GCF: Combining the GCF of the numerical coefficients and the variable 'r', we find that the GCF of 26r³s, 52r⁵, and 39r²s⁴ is 13r².

5. Factoring the Expression: To factor the expression, we divide each term by the GCF (13r²):

   • 26r³s / 13r² = 2rs
   • 52r⁵ / 13r² = 4r³
   • 39r²s⁴ / 13r² = 3s⁴

Therefore, the factored expression is 13r²(2rs + 4r³ + 3s⁴).

Advanced Factoring Techniques

While finding the GCF is a fundamental approach, other techniques can be employed for more complex expressions:

• Grouping: This method is useful when dealing with four or more terms. Group terms with common factors, then factor out the GCF from each group.

• Difference of Squares: This technique applies to expressions in the form a² - b², which factors as (a + b)(a - b).

• Perfect Square Trinomials: Expressions like a² + 2ab + b² factor as (a + b)².

• Sum and Difference of Cubes: These techniques involve factoring expressions of the form a³ + b³ and a³ - b³.

Applications of Factoring

Factoring is not just an abstract mathematical exercise; it has several practical applications:

• Simplifying Expressions: Factoring simplifies complex algebraic expressions, making them easier to manipulate and understand.

• Solving Equations: Factoring is crucial for solving polynomial equations. By setting each factor to zero, you can find the roots (solutions) of the equation.

• Calculus: Factoring is heavily used in calculus for simplifying derivatives and integrals.

• Real-World Problems: Factoring helps in solving real-world problems involving areas, volumes, and optimization.

Conclusion

Factoring algebraic expressions is a core skill in algebra and beyond. Mastering the techniques outlined in this article, starting with finding the greatest common factor, will equip you to tackle increasingly complex expressions. Remember to practice regularly to reinforce your understanding and improve your proficiency. By understanding GCF and applying various factoring techniques, you can efficiently simplify expressions, solve equations, and delve deeper into the world of mathematics. The process of factoring, while sometimes intricate, empowers you to break down complex expressions into their fundamental components, revealing underlying structures and patterns. The ability to factor effectively unlocks a deeper understanding of algebraic relationships and paves the way for tackling more advanced mathematical concepts.