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

Factor 26r3s 52r5 39r2s4 – What Is the Resulting Expression?

Introduction

When students first encounter algebraic expressions that contain several terms, the immediate question is often, “How can I simplify this?By breaking the problem into manageable steps, we will discover the greatest common factor (GCF) and rewrite the original sum as a product of simpler components. And ” In this article we will explore the process of factoring the expression 26r³s + 52r⁵ + 39r²s⁴. Understanding this technique not only reduces the complexity of the expression but also lays the groundwork for more advanced topics such as solving equations, simplifying rational expressions, and analyzing polynomial behavior Not complicated — just consistent. That alone is useful..

Understanding the Expression

The expression consists of three monomials:

1. 26r³s – a term with coefficient 26, a variable r raised to the third power, and a variable s to the first power.
2. 52r⁵ – a term with coefficient 52 and r raised to the fifth power (no s).
3. 39r²s⁴ – a term with coefficient 39, r squared, and s to the fourth power.

Each monomial can be broken down into a numeric factor and variable factors. Recognizing the structure of each term is the first step toward finding a common factor that can be pulled out It's one of those things that adds up. Turns out it matters..

Identifying Common Factors

Extracting the Numerical GCF

The coefficients are 26, 52, and 39.

• 26 = 2 × 13
• 52 = 4 × 13
• 39 = 3 × 13

The largest integer that divides all three numbers is 13. That's why, 13 is the numerical part of the GCF.

Extracting the Variable GCF

Now look at the variable parts:

• r³s contains r³ and s¹.
• r⁵ contains r⁵ (no s).
• r²s⁴ contains r² and s⁴.

The common variable factor must be present in all terms. Which means the smallest power of r that appears in every term is r², and there is no s common to all three (the first term has s¹, the second has none, the third has s⁴). This means the variable GCF is r².

Combining the numerical and variable parts, the overall GCF is 13r².

Step‑by‑Step Factoring Process

1. Pull Out the GCF

Write the original expression and factor out 13r²:

[ 26r^3s + 52r^5 + 39r^2s^4 = 13r^2 \bigl( ,?, \bigr) ]

To find what goes inside the parentheses, divide each original term by 13r²:

• ( \frac{26r^3s}{13r^2} = 2rs )
• ( \frac{52r^5}{13r^2} = 4r^3 )
• ( \frac{39r^2s^4}{13r^2} = 3s^4 )

Thus the factored form becomes:

[ 13r^2 \bigl( 2rs + 4r^3 + 3s^4 \bigr) ]

2. Examine the Remaining Polynomial

The expression inside the parentheses, 2rs + 4r³ + 3s⁴, still contains three terms. We now ask whether it can be factored further.

• Look for a common factor among 2rs, 4r³, and 3s⁴.
• The coefficients 2, 4, and 3 have no common divisor greater than 1.
• The variable parts share no common factor (r appears in the first two terms, s appears in the first and third).

That's why, 2rs + 4r³ + 3s⁴ is already in its simplest factored form.

3. Write the Final Result

The completely factored expression is:

[ \boxed{13r^2 \bigl( 2rs + 4r^3 + 3s^4 \bigr)} ]

This representation shows the product of the greatest common factor 13r² and the remaining simplified polynomial.

Scientific Explanation

Factoring is more than a mechanical manipulation; it reveals the structure of an algebraic object. In polynomial theory, a factor is a building block that, when multiplied by another factor, reproduces the original expression. By extracting the GCF, we isolate the largest piece that is common to all terms, which can simplify later steps such as solving equations (set the expression equal to zero) or reducing fractions (cancel common factors) Simple, but easy to overlook..

Beyond that, recognizing the GCF enhances computational efficiency. Here's one way to look at it: if we needed to evaluate the expression for specific values of r and s, having the factored form means we only need to compute 13r² once and then evaluate the simpler bracketed term, reducing the chance of arithmetic errors Easy to understand, harder to ignore..

Common Mistakes and How to Avoid Them

1. Skipping the Variable GCF – Some learners notice the numerical GCF (13) but forget that r² also appears in every term. Always check both numeric and variable parts.
2. Dividing Incorrectly – When extracting the GCF, a common error is to mis‑divide a term, leading to an incorrect bracket. Double‑check each division:
   • 26 ÷ 13 = 2, r³ ÷ r² = r, s ÷ 1 = s → 2rs (correct).
   • 52 ÷ 13 = 4, r⁵ ÷ r² = r³ → 4r³ (correct).
   • 39 ÷ 13 = 3, s⁴ ÷ 1 = s⁴ → 3s⁴ (correct).
3. Assuming Further Factorization – The bracket 2rs + 4r³ + 3s⁴ cannot be factored further using integer coefficients. Resist the urge to force a factor that does not exist; instead, verify by trying to regroup or factor by grouping, which often shows that no common factor remains.

FAQ

Q1: What does “factor” mean in algebra?
A: To factor an expression means to rewrite it as a product of two or more simpler expressions (factors). This often involves pulling out a greatest common factor or using special patterns like difference of squares.

Q2: Why is the GCF important?
A: The GCF is the largest piece that divides every term without leaving a remainder. Extracting it simplifies the expression and is the first step in many algebraic manipulations.

Q3: Can I factor out a variable that does not appear in every term?
A: No. A factor must be present in all terms. If a variable is missing from even one term, it cannot be part of the GCF.

Q4: Is the final answer unique?
A: Up to multiplication by a non‑zero constant, the factored form is unique. In our case, 13r² is the only GCF, and the remaining polynomial is already simplified Worth keeping that in mind. Practical, not theoretical..

Q5: How does factoring help solve equations?
A: Once an equation is set to zero, the factored form allows us to use the zero‑product property: if a product equals zero, at least one factor must be zero. This breaks a complex problem into simpler sub‑problems.

Conclusion

Factoring the expression 26r³s + 52r⁵ + 39r²s⁴ teaches us a systematic approach: identify the numerical and variable commonalities, extract the greatest common factor, and simplify the remaining terms. Think about it: the resulting expression, 13r² ( 2rs + 4r³ + 3s⁴ ), is cleaner, easier to work with, and highlights the underlying structure of the original polynomial. Mastering this technique equips students with a powerful tool for tackling more detailed algebraic problems, from solving equations to simplifying rational expressions. By practicing the steps outlined above, learners can confidently factor similar multi‑term expressions and appreciate the elegance of algebraic factorization.

Applying the Method to Similar Problems

The same strategy can be applied to any polynomial with multiple variables and coefficients. Here are a few additional examples to illustrate the process:

Example 1: $18x^2y + 24xy^2$

1. Identify the GCF:

   • Coefficients: $\gcd(18, 24) = 6$
   • Variables: Both terms contain at least $x$ and $y$, so the variable part is $xy$
   • GCF: $6xy$

2. Divide each term by the GCF:

   • $18x^2y \div 6xy = 3x$
   • $24xy^2 \div 6xy = 4y$

3. Write the factored form:
   $ 18x^2y + 24xy^2 = 6xy(3x + 4y) $

Example 2: $45a^3b^2c - 30a^2b^3c + 15abc$

1. Identify the GCF:

   • Coefficients: $\gcd(45, 30, 15) = 15$
   • Variables: All terms contain at least $a$, $b$, and $c$, so the variable part is $abc$
   • GCF: $15abc$

2. Divide each term by the GCF:

   • $45a^3b^2c \div 15abc = 3a^2b$
   • $-30a^2b^3c \div 15abc = -2ab^2$
   • $15abc \div 15abc = 1$

3. Write the factored form:
   $ 45a^3b^2c - 30a^2b^3c + 15abc = 15abc(3a^2b - 2ab^2 + 1) $

Tips for Success

• Always start with the coefficients: Find the greatest common divisor of all numerical terms first.
• Check every variable: Only include variables that appear in every term, raised to the lowest power found across all terms.
• Verify your divisions: After factoring out the GCF, multiply back to ensure you recover the original expression.
• Look ahead: Sometimes, factoring out a negative GCF or rearranging terms can make further factorization possible.

Conclusion

Factoring expressions like $26r^3s + 52r^5 + 39r^2s^4$ reinforces foundational skills essential for advanced mathematics. By identifying the greatest common factor—here, $13r^2$—and simplifying each term accordingly, we arrive at a more manageable form:
$ 13r^2(2rs + 4r^3 + 3s^4) $
This process not only streamlines calculations but also reveals the structural relationships within algebraic expressions. Whether solving equations, simplifying fractions, or preparing for higher-level topics like polynomial division, mastering GCF-based factoring is indispensable. Through consistent practice and attention to detail, students develop both accuracy and confidence in their algebraic reasoning.