Factoring 26r³s + 52r⁵ - 39r²s⁴ A Step-by-Step Guide

Factoring algebraic expressions is a fundamental skill in mathematics, particularly in algebra. It involves breaking down an expression into simpler terms, usually by identifying common factors. This not only simplifies the expression but also makes it easier to solve equations and analyze mathematical relationships. In this article, we will delve into the process of factoring the expression 26r³s + 52r⁵ - 39r²s⁴, providing a step-by-step guide to understanding and solving it. This detailed explanation will help students, educators, and anyone interested in mathematics to grasp the nuances of factoring and apply these techniques to various algebraic problems. Understanding the techniques used here will serve as a cornerstone for tackling more complex algebraic manipulations and problem-solving scenarios.

Identifying Common Factors

The first step in factoring any algebraic expression is to identify the common factors among the terms. Common factors are terms that divide evenly into each part of the expression. In the given expression, 26r³s + 52r⁵ - 39r²s⁴, we need to look for both numerical and variable factors that are common to all terms. This involves carefully examining the coefficients (the numbers in front of the variables) and the variables themselves, noting their exponents. The goal is to find the greatest common factor (GCF), which is the largest factor that all terms share. Identifying the GCF correctly is crucial as it allows us to simplify the expression most effectively. Failing to identify the complete GCF may result in partially factored expressions, which, while mathematically correct, do not fully simplify the original expression. This step is not just about finding any common factor but about finding the greatest one, ensuring the expression is factored to its simplest form.

Numerical Factors

When examining the numerical factors, we look at the coefficients: 26, 52, and -39. We need to find the largest number that divides all three coefficients without leaving a remainder. To do this, we can list the factors of each number:

• Factors of 26: 1, 2, 13, 26
• Factors of 52: 1, 2, 4, 13, 26, 52
• Factors of -39: 1, 3, 13, 39

By comparing these lists, we can see that the greatest common numerical factor is 13. This means that 13 is the largest number that divides 26, 52, and -39 evenly. Recognizing this numerical GCF is a critical step, as it allows us to simplify the coefficients in the original expression significantly. This simplification is not just a matter of reducing the size of the numbers; it also helps in revealing the underlying structure of the expression, making further factorization steps more manageable and intuitive. The numerical GCF essentially sets the scale for how much we can initially simplify the entire expression.

Variable Factors

Next, we consider the variable factors in the expression 26r³s + 52r⁵ - 39r²s⁴. We look for the variables that are present in each term, which are r and s. For each variable, we take the lowest exponent that appears in any term. This is because the lowest exponent indicates the maximum power of the variable that can be factored out from all terms. For r, the exponents are 3, 5, and 2. The lowest exponent is 2, so we can factor out r². For s, the exponents are 1 (in the first term) and 4 (in the third term). The lowest exponent is 1, so we can factor out s. Therefore, the greatest common variable factor is r²s. This careful selection of the lowest exponents ensures that we are not trying to factor out more of a variable than is available in any of the terms, which is a common mistake in factoring. The process of identifying variable factors is crucial for simplifying expressions and preparing them for further algebraic manipulations, such as solving equations or simplifying rational expressions. Correctly identifying and factoring out variable factors streamlines the entire process, making it less prone to errors.

Combining Numerical and Variable Factors

Having identified both the greatest common numerical factor (13) and the greatest common variable factor (r²s), we can now combine them to find the overall greatest common factor (GCF) of the expression 26r³s + 52r⁵ - 39r²s⁴. By multiplying the numerical and variable GCFs, we get 13r²s. This means that 13r²s is the largest expression that divides evenly into each term of the original expression. Recognizing this overall GCF is the cornerstone of the factoring process. It is the key to simplifying the expression in the most efficient way possible. This combined GCF represents the maximum simplification achievable in the first step of factoring, and it sets the stage for subsequent steps, if any are needed. Accurately determining this combined GCF is a critical skill in algebra, and it is essential for anyone looking to master factoring and related algebraic manipulations.

Factoring Out the GCF

Now that we have determined the greatest common factor (GCF) to be 13r²s, the next step is to factor it out of the original expression, 26r³s + 52r⁵ - 39r²s⁴. Factoring out the GCF involves dividing each term in the expression by the GCF and writing the expression as a product of the GCF and the resulting quotient. This process is essentially the reverse of distribution, and it is a crucial step in simplifying algebraic expressions. Factoring out the GCF not only reduces the complexity of the expression but also reveals the underlying structure, making it easier to analyze and manipulate. This step is particularly useful in solving equations, simplifying rational expressions, and identifying key properties of algebraic functions. Successfully factoring out the GCF sets the stage for further simplification or problem-solving steps.

Dividing Each Term by the GCF

To factor out the GCF, 13r²s, we divide each term of the expression 26r³s + 52r⁵ - 39r²s⁴ by 13r²s. This involves dividing the numerical coefficients and applying the rules of exponents for the variables. Let's break it down term by term:

1. 26r³s / (13r²s):
   • Divide the coefficients: 26 / 13 = 2
   • Divide the variables: r³ / r² = r^(3-2) = r; s / s = 1
   • Result: 2r
2. 52r⁵ / (13r²s):
   • Divide the coefficients: 52 / 13 = 4
   • Divide the variables: r⁵ / r² = r^(5-2) = r³; since there is no 's' in the denominator to cancel out the 's' in the numerator, we consider this term as having s⁰, thus s⁰ = 1, so there's effectively no 's' term in the result.
   • Result: 4r³
3. -39r²s⁴ / (13r²s):
   • Divide the coefficients: -39 / 13 = -3
   • Divide the variables: r² / r² = 1; s⁴ / s = s^(4-1) = s³
   • Result: -3s³

This careful division of each term by the GCF is a critical step, as it ensures that the factored expression is mathematically equivalent to the original expression. Each division should be double-checked to avoid errors, as any mistake here will propagate through the rest of the factoring process. The results of these divisions form the terms inside the parentheses in the factored expression.

Writing the Factored Expression

After dividing each term by the GCF, we can now write the factored expression. We write the GCF, 13r²s, outside the parentheses and the results of the division inside the parentheses. So, the factored expression is:

**13r²s(2r + 4r³ - 3s³) **

This resulting expression is a product of the GCF and the simplified terms within the parentheses. This form highlights the common factor that was present in all terms of the original expression, and it represents a simplified way of expressing the same mathematical relationship. Writing the factored expression correctly is essential, as it is the final result of the factoring process. It's always a good practice to mentally distribute the GCF back into the parentheses to ensure that the factored expression is indeed equivalent to the original expression. This check helps to catch any errors made during the division process and confirms the correctness of the factoring.

Final Answer

Therefore, the factored form of the expression 26r³s + 52r⁵ - 39r²s⁴ is **13r²s(2r + 4r³ - 3s³) **. This matches option B in the given choices. This final factored form represents the simplified expression, and it is often easier to work with in various mathematical contexts. The process of factoring not only simplifies the expression but also provides insights into its structure and properties. This is particularly useful in solving equations, simplifying rational expressions, and analyzing algebraic functions. The ability to factor expressions efficiently and accurately is a fundamental skill in algebra, and it forms the basis for more advanced mathematical concepts.

In summary, factoring involves several key steps: identifying the greatest common factor (GCF), dividing each term by the GCF, and writing the expression as a product of the GCF and the resulting quotient. By carefully applying these steps, we can simplify complex algebraic expressions and gain a deeper understanding of their mathematical properties.

Choosing the Correct Option

After successfully factoring the expression 26r³s + 52r⁵ - 39r²s⁴ into **13r²s(2r + 4r³ - 3s³) **, we can confidently choose the correct option from the given choices. The factored expression clearly matches option B:

**B. 13r²s(2r + 4r³ - 3s³) **

Options A, C, and D are incorrect because they do not represent the fully and correctly factored form of the original expression. Option A, 13(2r³s + 4r⁵ - 3r²s⁴), only factors out the numerical factor 13 but misses the variable factors. Option C, 13r²(2rs + 4r³ - 3s⁴), correctly factors out 13r², but it misses the 's' term that should be factored out from each term. Option D, 13r²(26r³s + 52r⁵ - 39r²s⁴), is incorrect as it does not simplify the expression at all; it seems to multiply some terms by 13r² instead of factoring. Therefore, carefully comparing our derived factored expression with the given options leads us to the unambiguous conclusion that option B is the correct answer. This final step of choosing the correct option underscores the importance of accuracy in each step of the factoring process, as any mistake along the way could lead to selecting the wrong answer.