Factoring Polynomials: Step-by-Step Solution

Aug 6, 2025 by Rajiv Sharma 45 views

Factoring Polynomials: A Step-by-Step Solution for 36r^9 - 9r^6 + 54r^3

Hey everyone! Let's dive into the world of factoring polynomials. Factoring polynomials can seem daunting, but with a systematic approach, it becomes much more manageable. In this article, we'll break down the process of factoring the polynomial 36r^9 - 9r^6 + 54r^3 step by step. We'll cover identifying the greatest common factor (GCF), factoring it out, and ensuring our final answer is in the simplest form. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into solving, let's clearly understand the problem. We're given the polynomial 36r^9 - 9r^6 + 54r^3 and our goal is to find its factored form. Factoring involves breaking down a polynomial into simpler expressions that, when multiplied together, give us the original polynomial. Think of it like reversing the distributive property.

To kick things off, let's analyze the coefficients and variables in our polynomial:

Coefficients: We have 36, -9, and 54.
Variables: We have r raised to the powers of 9, 6, and 3 (r^9, r^6, r^3).

Our strategy will be to first identify the greatest common factor (GCF) of both the coefficients and the variable terms. Once we find the GCF, we'll factor it out, leaving us with a simplified expression inside the parentheses. This is a crucial step in factoring any polynomial, and it sets the stage for further simplification if needed.

Finding the Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is the cornerstone of factoring polynomials. It's like finding the biggest piece that fits into all the terms of our polynomial. To find the GCF, we look at the coefficients (the numbers) and the variables separately.

GCF of the Coefficients

First, let's tackle the coefficients: 36, -9, and 54. We need to find the largest number that divides evenly into all three. Here’s how we can break it down:

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of -9: -1, -3, -9, 1, 3, 9
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54

By looking at these factors, we can see that the greatest common factor of 36, -9, and 54 is 9. This means 9 is the largest number that can divide all three coefficients without leaving a remainder. Keep this in mind as we move to the next step, which involves finding the GCF of the variable terms.

GCF of the Variables

Next, we consider the variable terms: r^9, r^6, and r^3. When finding the GCF of variables with exponents, we take the variable with the smallest exponent. In this case, we have r raised to the powers of 9, 6, and 3. The smallest exponent is 3, so the GCF of the variable terms is r^3.

Think of it this way: r^3 is the highest power of r that is present in all three terms. We can rewrite each term to show this:

36r^9 = 36 * r^3 * r^6
-9r^6 = -9 * r^3 * r^3
54r^3 = 54 * r^3

This makes it clear that r^3 is a common factor.

Combining the GCFs

Now that we've found the GCF of the coefficients (9) and the GCF of the variables (r^3), we can combine them to find the overall GCF of the polynomial. The overall GCF is simply the product of these two GCFs, which is 9r^3. This means 9r^3 is the largest expression that can divide evenly into each term of the polynomial 36r^9 - 9r^6 + 54r^3.

Identifying the GCF is a crucial step because it allows us to simplify the polynomial by factoring it out. This not only makes the polynomial easier to work with but also helps us identify the correct factored form from the given options. In the next section, we'll use this GCF to factor the polynomial and see which of the provided choices matches our result. So, stick around, and let's continue factoring!

Factoring Out the GCF

Now that we've identified the greatest common factor (GCF) as 9r^3, it's time to factor it out of the polynomial 36r^9 - 9r^6 + 54r^3. Factoring out the GCF is like dividing each term of the polynomial by the GCF and then writing the GCF outside a set of parentheses. This process simplifies the polynomial and helps us get closer to the final factored form.

To factor out 9r^3, we'll divide each term of the polynomial by 9r^3 and write the result inside the parentheses:

Divide 36r^9 by 9r^3:
- (36r^9) / (9r^3) = (36/9) * (r^9/r3) = 4r^(9-3) = 4r^6
Divide -9r^6 by 9r^3:
- (-9r^6) / (9r^3) = (-9/9) * (r^6/r3) = -1r^(6-3) = -r^3
Divide 54r^3 by 9r^3:
- (54r^3) / (9r^3) = (54/9) * (r^3/r3) = 6r^(3-3) = 6r^0 = 6 * 1 = 6

Now, we write the GCF outside the parentheses and the results of our divisions inside the parentheses. This gives us:

9r^3(4r6 - r^3 + 6)

This is the factored form of the polynomial 36r^9 - 9r^6 + 54r^3 after factoring out the GCF. We've essentially reversed the distributive property. If we were to distribute 9r^3 back into the parentheses, we would get the original polynomial.

Factoring out the GCF is a powerful technique. It not only simplifies the polynomial but also makes it easier to identify further factoring opportunities. In this case, the expression inside the parentheses (4r^6 - r^3 + 6) doesn't have any common factors, and it's not a quadratic form that we can easily factor further. Therefore, our factored form is complete.

In the next section, we'll compare our factored form with the given options to identify the correct answer. This step is crucial to ensure we've not only factored correctly but also presented our answer in the format required by the question. Let's move on and nail this problem!

Comparing with the Given Options

Alright, we've successfully factored the polynomial 36r^9 - 9r^6 + 54r^3 and arrived at the factored form: 9r^3(4r6 - r^3 + 6). Now, the crucial step is to compare our result with the given options to identify the correct answer. This ensures that we've not only performed the factoring correctly but also presented the answer in the required format.

Let's take a look at the options provided:

A. 9r^3(4r3 - 9r^6 + 54r^3) B. 9r^3(4r6 - 9r^6 + 54r^3) C. 9r^3(4r3 - r^2 + 6) D. 9r^3(4r6 - r^3 + 6)

Now, let's compare each option with our factored form:

Option A: 9r^3(4r3 - 9r^6 + 54r^3)
- This option doesn't match our factored form. The terms inside the parentheses are different.
Option B: 9r^3(4r6 - 9r^6 + 54r^3)
- This option also doesn't match our factored form. The terms inside the parentheses are incorrect.
Option C: 9r^3(4r3 - r^2 + 6)
- This option is close, but it's not quite right. The terms inside the parentheses don't match our result.
Option D: 9r^3(4r6 - r^3 + 6)
- This option perfectly matches our factored form! The GCF 9r^3 is factored out correctly, and the terms inside the parentheses (4r^6 - r^3 + 6) are exactly what we obtained.

By carefully comparing our factored form with the given options, we can confidently conclude that Option D is the correct answer. This methodical approach is essential to avoid errors and ensure we select the right solution.

Conclusion

In this article, we walked through the process of factoring the polynomial 36r^9 - 9r^6 + 54r^3. We started by understanding the problem and identifying the greatest common factor (GCF), which was 9r^3. Then, we factored out the GCF, resulting in the expression 9r^3(4r6 - r^3 + 6). Finally, we compared our factored form with the given options and found that Option D was the correct answer.

Factoring polynomials is a fundamental skill in algebra, and mastering it requires practice and a systematic approach. Remember to always look for the GCF first, factor it out, and then check if the remaining expression can be factored further. By following these steps, you'll be well-equipped to tackle a wide range of factoring problems.

So, keep practicing, and you'll become a factoring pro in no time! If you have any questions or want to explore more factoring techniques, feel free to ask. Happy factoring, guys!