Factoring Polynomials By Grouping A Comprehensive Guide To X^3 - 9x^2 + 5x - 45

Jul 15, 2025 by ADMIN 80 views

Factoring polynomials can sometimes feel like solving a puzzle, especially when dealing with cubic expressions. One effective technique for tackling such polynomials is factoring by grouping. This method involves strategically grouping terms, identifying common factors, and then extracting those factors to simplify the expression. In this article, we will delve into the process of factoring the polynomial $x^3 - 9x^2 + 5x - 45$ using the grouping method, while also dissecting the multiple-choice options to pinpoint the correct approach. This detailed explanation aims to provide a clear understanding of the underlying principles and steps involved in polynomial factorization.

Understanding Factoring by Grouping

Factoring by grouping is a powerful technique used to simplify polynomials, especially those with four or more terms. The core idea behind this method is to identify common factors within smaller groups of terms and then extract these factors to reveal a shared binomial or polynomial factor. This shared factor can then be factored out, leading to a simplified expression. This method is particularly useful when dealing with cubic polynomials or higher-degree polynomials that do not readily fit into standard factoring patterns.

The general steps involved in factoring by grouping are as follows:

Group the terms: Arrange the terms of the polynomial into pairs or groups. The grouping should be done in such a way that each group has a common factor.
Factor out the greatest common factor (GCF) from each group: Identify the GCF in each group and factor it out. This step should ideally result in each group having the same binomial factor.
Factor out the common binomial: If the groups now share a common binomial factor, factor it out from the entire expression. This will leave you with a product of two factors.
Check for further factoring: After factoring, always check if the resulting factors can be further factored. This ensures that the polynomial is completely factored.

By following these steps, you can effectively factor a wide range of polynomials that might initially seem complex. The key is to look for patterns and common factors that will allow you to simplify the expression.

Factoring x^3 - 9x^2 + 5x - 45 by Grouping

Let’s apply the factoring by grouping method to the polynomial $x^3 - 9x^2 + 5x - 45$ . This polynomial has four terms, making it a prime candidate for this technique. The goal is to rewrite the polynomial as a product of simpler expressions.

Step 1: Group the Terms

The first step in factoring by grouping is to group the terms in a way that allows us to identify common factors. A natural grouping for this polynomial is to pair the first two terms and the last two terms:

$(x^3 - 9x^2) + (5x - 45)$

This grouping is strategic because the first group contains terms with $x$ as a factor, while the second group contains terms with a numerical factor. This sets the stage for the next step, where we factor out the greatest common factor from each group.

Step 2: Factor out the GCF from Each Group

Now, we identify the greatest common factor (GCF) in each group and factor it out.

In the first group, $(x^3 - 9x^2)$ , the GCF is $x^2$ . Factoring out $x^2$ gives us: $x^2(x - 9)$
In the second group, $(5x - 45)$ , the GCF is $5$ . Factoring out $5$ gives us: $5(x - 9)$

After factoring out the GCF from each group, our expression looks like this:

$x^2(x - 9) + 5(x - 9)$

Notice that both groups now share a common binomial factor: $(x - 9)$ . This is a crucial step in the factoring by grouping process. The presence of a common binomial factor indicates that we are on the right track.

Step 3: Factor out the Common Binomial

Since both groups share the binomial factor $(x - 9)$ , we can factor it out from the entire expression. This is similar to factoring out a GCF, but in this case, the GCF is a binomial.

Factoring out $(x - 9)$ from $x^2(x - 9) + 5(x - 9)$ gives us:

$(x - 9)(x^2 + 5)$

This is the factored form of the polynomial $x^3 - 9x^2 + 5x - 45$ . We have successfully rewritten the cubic polynomial as a product of two factors: a binomial $(x - 9)$ and a quadratic $(x^2 + 5)$ .

Step 4: Check for Further Factoring

As a final step, we should check if either of the factors can be further factored. The binomial factor $(x - 9)$ is a simple linear term and cannot be factored further. The quadratic factor $(x^2 + 5)$ is a sum of squares, and since there are no real numbers that satisfy the condition for factoring a sum of squares, it cannot be factored further using real numbers.

Thus, the completely factored form of the polynomial $x^3 - 9x^2 + 5x - 45$ is $(x - 9)(x^2 + 5)$ .

Analyzing the Multiple-Choice Options

Now that we have factored the polynomial $x^3 - 9x^2 + 5x - 45$ by grouping, let's analyze the multiple-choice options provided to identify the one that correctly shows the intermediate step in the factoring process.

The multiple-choice options are:

A. $x^2(x - 9) - 5(x - 9)$
B. $x^2(x + 9) - 5(x + 9)$
C. $x(x^2 + 5) - 9(x^2 + 5)$
D. $x(x^2 - 5) - 9(x^2 - 5)$

Comparing these options with the steps we took to factor the polynomial, we can see that option A, $x^2(x - 9) + 5(x - 9)$ , closely matches one of our intermediate steps. However, there's a small difference in the sign of the second term. Let's re-examine our steps to ensure we have the correct sign.

Going back to Step 2, where we factored out the GCF from each group, we had:

$x^2(x - 9) + 5(x - 9)$

This matches option A closely, but option A has a minus sign instead of a plus sign. This indicates a potential error in the original options or a slight modification needed to make it correct. Option A should be $x^2(x - 9) + 5(x - 9)$ . However, given the options, we need to find the closest correct representation.

Looking at option C, $x(x^2 + 5) - 9(x^2 + 5)$ , we can see that if we factor out the common binomial $(x^2 + 5)$ , we get:

$(x^2 + 5)(x - 9)$

This is the factored form we derived earlier, but the intermediate step presented in option C is not a direct result of the initial grouping we performed. Instead, it's a step that would occur if we rearranged the terms differently.

Option B, $x^2(x + 9) - 5(x + 9)$ , is incorrect because it includes the term $(x + 9)$ , which is not a factor we encountered during our factoring process. Similarly, option D, $x(x^2 - 5) - 9(x^2 - 5)$ , is also incorrect as $(x^2 - 5)$ was not a factor we identified.

Therefore, the closest option that shows one way to determine the factors by grouping, even with the sign discrepancy, is option A if we consider a minor correction. However, option C presents a valid intermediate step if the terms were rearranged, making it a plausible answer as well.

Conclusion

Factoring polynomials by grouping is a valuable technique for simplifying expressions and solving algebraic problems. In the case of the polynomial $x^3 - 9x^2 + 5x - 45$ , we demonstrated how to group terms, factor out common factors, and arrive at the factored form $(x - 9)(x^2 + 5)$ .

Analyzing the multiple-choice options, we identified that option A, $x^2(x - 9) + 5(x - 9)$ , represents a key intermediate step in the factoring process, although it had a sign discrepancy in the original question. Option C, $x(x^2 + 5) - 9(x^2 + 5)$ , also represents a valid intermediate step if the terms were rearranged differently.

By understanding the principles of factoring by grouping and practicing various examples, you can develop the skills needed to tackle more complex polynomial expressions. This method not only simplifies expressions but also provides insights into the structure and properties of polynomials.

Keywords: Factoring polynomials, factoring by grouping, polynomial factorization, cubic polynomials, greatest common factor, binomial factor, algebraic expressions, mathematics, $x^3 - 9x^2 + 5x - 45$ , intermediate steps.