Which Option Correctly Demonstrates The Initial Step In Determining The Factors Of The Polynomial X³ - 9x² + 5x - 45 By Grouping?

Jun 19, 2025 by ADMIN 130 views

Factoring by Grouping An In-Depth Guide to Determining Factors of x³ - 9x² + 5x - 45

Factoring polynomials is a fundamental skill in algebra, and one powerful technique for achieving this is factoring by grouping. This method is particularly useful when dealing with polynomials that have four or more terms. In this comprehensive guide, we will explore the process of factoring by grouping, using the specific example of the polynomial x³ - 9x² + 5x - 45. We'll dissect the steps involved, explain the underlying principles, and provide clear explanations to help you master this technique.

Understanding Factoring by Grouping

Factoring by grouping is a strategy that allows us to break down a polynomial into simpler factors by strategically grouping terms together. The main idea behind factoring by grouping is to identify common factors within groups of terms and then factor those common factors out. This process, when successful, results in a common binomial factor that can be factored out from the entire polynomial.

Before we dive into the specifics of our example, let's outline the general steps involved in factoring by grouping:

Group Terms: Arrange the terms of the polynomial into pairs. Look for pairs that might have a common factor.
Factor Each Group: Factor out the greatest common factor (GCF) from each pair of terms.
Identify Common Binomial Factor: If the factoring is successful, you should now have two terms, each containing the same binomial factor.
Factor out the Binomial Factor: Factor out the common binomial factor from the entire expression.
Final Factored Form: You should now have the polynomial expressed as a product of two factors.

Now, let's apply these steps to our specific polynomial: x³ - 9x² + 5x - 45.

Step-by-Step Factoring of x³ - 9x² + 5x - 45

1. Grouping the Terms

The first step in factoring by grouping is to pair up the terms in the polynomial. When choosing which terms to group, look for pairs that share a common factor. In our case, the polynomial is x³ - 9x² + 5x - 45. A natural grouping here is to pair the first two terms and the last two terms:

(x³ - 9x²) + (5x - 45)

This grouping makes sense because x³ and -9x² both have x² as a common factor, and 5x and -45 both have 5 as a common factor. Strategic grouping is essential for the success of this method.

2. Factoring Each Group

Next, we factor out the greatest common factor (GCF) from each group. In the first group, (x³ - 9x²), the GCF is x². Factoring out x² gives us:

x²(x - 9)

In the second group, (5x - 45), the GCF is 5. Factoring out 5 gives us:

5(x - 9)

Now, our expression looks like this:

x²(x - 9) + 5(x - 9)

Notice that we now have two terms, and both terms contain the same binomial factor: (x - 9). This is a crucial step in factoring by grouping.

3. Identifying the Common Binomial Factor

The key to factoring by grouping lies in identifying the common binomial factor. In our expression, x²(x - 9) + 5(x - 9), the common binomial factor is clearly (x - 9). This common factor is what allows us to proceed with the next step.

4. Factoring out the Binomial Factor

Now that we've identified the common binomial factor, we can factor it out from the entire expression. We treat (x - 9) as a single term and factor it out, just like we would factor out a single variable or constant. This gives us:

(x - 9)(x² + 5)

We have now factored out the common binomial factor, and we are one step closer to the final factored form of the polynomial.

5. Final Factored Form

We have successfully factored the polynomial by grouping. Our final factored form is:

(x - 9)(x² + 5)

This is the completely factored form of the polynomial x³ - 9x² + 5x - 45. We have expressed the original polynomial as a product of two factors: (x - 9) and (x² + 5).

Analyzing the Given Options

Now that we have factored the polynomial x³ - 9x² + 5x - 45 by grouping, let's analyze the given options to see which one demonstrates the correct first step in this process.

A. x²(x - 9) - 5(x - 9)

This option is close to the correct factoring but has a sign error. If we distribute the -5, we would get -5x + 45, whereas the original polynomial has +5x - 45. Thus, this is not the correct grouping.

B. x²(x + 9) - 5(x + 9)

This option is incorrect because the binomial factor (x + 9) does not appear when factoring the original polynomial. The signs are incorrect, and this grouping does not lead to the correct factored form.

C. x²(x - 9) + 5(x - 9)

This option accurately shows the correct grouping and factoring of the GCF from each pair of terms. As we demonstrated in our step-by-step solution, this is the correct intermediate step in factoring the polynomial by grouping. The common binomial factor (x - 9) is clearly visible.

D. x(x² - 5) - 9(x² - 5)

This option is also incorrect. While it demonstrates an attempt to factor by grouping, the resulting binomial factor (x² - 5) does not align with the correct factoring process for this polynomial. If this were the correct grouping, we could then factor out (x² - 5) to get (x - 9)(x² - 5), which would yield a different polynomial than the one we started with.

Therefore, the correct option is C, as it accurately shows the intermediate step in factoring x³ - 9x² + 5x - 45 by grouping.

Tips and Tricks for Factoring by Grouping

Factoring by grouping can be a challenging technique, but with practice and a few helpful tips, you can master it. Here are some key strategies to keep in mind:

Look for Common Factors: Always start by looking for the greatest common factor (GCF) in each group of terms. Factoring out the GCF is crucial for revealing the common binomial factor.
Pay Attention to Signs: Be meticulous about signs. A small sign error can throw off the entire factoring process. Ensure that the signs within the binomial factors match up correctly.
Rearrange Terms if Necessary: Sometimes, the initial grouping of terms may not lead to a common binomial factor. In such cases, try rearranging the terms to find a more suitable grouping.
Check Your Work: After factoring, always multiply the factors back together to verify that you get the original polynomial. This is a great way to catch any errors.
Practice Regularly: Like any mathematical skill, factoring by grouping requires practice. The more you practice, the more comfortable you will become with the technique.

Common Mistakes to Avoid

When factoring by grouping, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your factoring skills.

Incorrect Grouping: Grouping terms that do not share a common factor can lead to failure. Make sure that the terms you group together have a GCF.
Sign Errors: As mentioned earlier, sign errors are a common issue. Double-check your signs when factoring out GCFs and binomial factors.
Forgetting to Factor Completely: After factoring out the binomial factor, make sure that the remaining factors cannot be factored further. Always aim for the completely factored form.
Skipping Steps: It can be tempting to skip steps in the factoring process, especially as you become more proficient. However, skipping steps increases the risk of making errors. Take your time and write out each step clearly.

Factoring by Grouping in Real-World Applications

Factoring polynomials, including factoring by grouping, is not just an abstract mathematical exercise. It has numerous applications in various real-world scenarios. Here are a few examples:

Engineering: Engineers use factoring to solve equations related to structural design, electrical circuits, and fluid dynamics.
Physics: Factoring is essential in physics for solving problems related to motion, energy, and quantum mechanics.
Computer Science: Factoring plays a role in algorithms for data compression, cryptography, and computer graphics.
Economics: Economists use factoring in mathematical models for economic forecasting and analysis.
Finance: Financial analysts use factoring to analyze investment portfolios and manage risk.

The ability to factor polynomials is a valuable skill that extends beyond the classroom, making it an important tool for problem-solving in various disciplines.

Conclusion

In this comprehensive guide, we have explored the technique of factoring by grouping, using the specific example of the polynomial x³ - 9x² + 5x - 45. We have dissected the steps involved, explained the underlying principles, and provided clear explanations to help you master this technique. Factoring by grouping is a powerful tool for simplifying polynomials and solving algebraic equations. By following the steps outlined in this guide and practicing regularly, you can become proficient in factoring by grouping and apply this skill to various mathematical and real-world problems. Remember to look for common factors, pay attention to signs, and always check your work to ensure accuracy. With dedication and practice, you can confidently tackle factoring problems and enhance your algebraic abilities.

In summary, the correct way to determine the factors of x³ - 9x² + 5x - 45 by grouping is demonstrated in option C: x²(x - 9) + 5(x - 9). This option accurately reflects the initial steps in the factoring by grouping process, setting the stage for the final factorization.