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

Factoring Expressions: A Deep Dive into 26r³s, 52r⁵, and 39r²s⁴

Finding the greatest common factor (GCF) of algebraic expressions is a fundamental skill in algebra. This process simplifies expressions, making them easier to manipulate and solve equations. This article will guide you through factoring the expressions 26r³s, 52r⁵, and 39r²s⁴, explaining the steps involved and providing a deeper understanding of the underlying principles. We'll explore numerical and variable factors, offering a comprehensive approach applicable to a wider range of similar problems.

Understanding the Components: Numerical and Variable Factors

Before diving into the factoring process, let's break down each expression into its constituent parts: numerical coefficients and variable factors.

• 26r³s: This expression consists of the numerical coefficient 26, the variable 'r' raised to the power of 3 (r³), and the variable 's' raised to the power of 1 (implicitly).

• 52r⁵: This expression comprises the numerical coefficient 52 and the variable 'r' raised to the power of 5 (r⁵).

• 39r²s⁴: This expression contains the numerical coefficient 39, the variable 'r' raised to the power of 2 (r²), and the variable 's' raised to the power of 4 (s⁴).

Finding the Greatest Common Factor (GCF)

The goal is to identify the largest factor common to all three expressions. We'll tackle this in two stages: finding the GCF of the numerical coefficients and then the GCF of the variable factors.

Finding the GCF of the Numerical Coefficients (26, 52, and 39)

To find the GCF of 26, 52, and 39, we'll use prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves).

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

By comparing the prime factorizations, we see that the only common prime factor among 26, 52, and 39 is 13. Therefore, the GCF of the numerical coefficients is 13.

Finding the GCF of the Variable Factors (r³s, r⁵, and r²s⁴)

Now, let's analyze the variable factors. We have 'r' and 's' in various powers. To find the GCF of the variable factors, we select the lowest power of each variable present in all expressions.

• r: The lowest power of 'r' among r³, r⁵, and r² is r².
• s: The variable 's' is present in only two of the three expressions (26r³s and 39r²s⁴). Therefore, we can only include the lowest power of 's' present in both of these expressions which is s¹.

Therefore, the GCF of the variable factors is r²s.

Combining the GCFs to Factor the Expressions

Now that we've determined the GCF of the numerical coefficients (13) and the variable factors (r²s), we can combine them to find the overall GCF of the three expressions. This is simply the product of the individual GCFs:

GCF = 13r²s

This means that 13r²s is the largest expression that divides evenly into 26r³s, 52r⁵, and 39r²s⁴.

Factoring the Expressions using the GCF

To complete the factoring, we divide each original expression by the GCF (13r²s):

• 26r³s / 13r²s = 2r
• 52r⁵ / 13r²s = 4r³/s
• 39r²s⁴ / 13r²s = 3s³

Therefore, the factored form of the expressions is:

**13r²s(2r + 4r³/s + 3s³) **

Advanced Factoring Techniques and Considerations

While the above method provides the complete factorization for the given expressions, understanding additional factoring techniques can prove valuable in tackling more complex algebraic problems.

Factoring Out Common Binomials

In more complex scenarios, you might encounter expressions with common binomial factors (expressions containing two terms). For example:

x(a + b) + y(a + b)

Here, (a + b) is a common binomial factor, and the expression can be factored as:

(a + b)(x + y)

This technique, while not applicable to our initial problem, showcases the broader applicability of factoring principles.

Factoring Trinomials

Trinomials (expressions with three terms) often require more advanced techniques. Methods like factoring by grouping and using the quadratic formula can be employed depending on the structure of the trinomial. For example:

x² + 5x + 6 can be factored as (x + 2)(x + 3).

Mastering these techniques broadens your ability to simplify and manipulate complex algebraic expressions.

Recognizing Special Cases

Certain patterns in algebraic expressions allow for quick factoring. These include:

• Difference of Squares: a² - b² = (a + b)(a - b)
• Perfect Square Trinomial: a² + 2ab + b² = (a + b)²
• Sum/Difference of Cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²)

Recognizing these patterns simplifies the factoring process considerably.

Applications of Factoring in Algebra and Beyond

Factoring is not merely an abstract algebraic exercise; it has numerous practical applications:

• Solving Quadratic Equations: Factoring is a key step in solving many quadratic equations, allowing you to find the roots (solutions) of the equation.

• Simplifying Rational Expressions: Factoring simplifies rational expressions (fractions with algebraic expressions in the numerator and denominator), allowing for cancellation of common factors.

• Calculus: Factoring plays a crucial role in calculus, particularly in differentiation and integration.

• Engineering and Physics: Many physical phenomena are modeled using algebraic equations, and factoring aids in analysis and problem-solving.

Conclusion

Factoring algebraic expressions is a fundamental skill with far-reaching applications. By understanding the steps involved in finding the greatest common factor and applying various factoring techniques, you can simplify complex expressions, solve equations, and progress in your mathematical studies. Remember to always break down the problem into its numerical and variable components, carefully analyze the factors, and utilize the appropriate factoring method to achieve the most simplified and accurate result. The process demonstrated with 26r³s, 52r⁵, and 39r²s⁴ provides a solid foundation for tackling more advanced factoring challenges.