Step-by-Step Factoring Of Polynomial 𝑏² + 3𝑏 − 𝑏⁴ − 3𝑏³ / 𝑏² − 9

by ADMIN 70 views

Factoring polynomials can seem daunting at first, but with a systematic approach, even complex expressions can be simplified. In this comprehensive guide, we'll break down the step-by-step process of factoring the polynomial b² + 3b − b⁴ − 3b³ / b² − 9. We'll walk through each stage, explaining the reasoning behind each step and providing helpful tips to make the process clearer. By the end of this guide, you'll have a solid understanding of how to factor this specific polynomial and be better equipped to tackle similar problems in the future. So, let’s dive in and conquer this factoring challenge together!

1. Understanding Polynomial Factoring

Before we jump into the specific polynomial, let's take a moment to understand the basics of polynomial factoring. Guys, think of factoring as the reverse of expanding. When we expand, we multiply terms together to get a larger expression. Factoring, on the other hand, involves breaking down a polynomial into its constituent factors – expressions that, when multiplied together, give us the original polynomial.

Polynomial factoring is a crucial skill in algebra and is used extensively in solving equations, simplifying expressions, and analyzing functions. There are several different techniques for factoring polynomials, and the best approach often depends on the specific form of the polynomial you're working with. Some common techniques include:

  • Greatest Common Factor (GCF): This involves identifying the largest factor that divides all terms in the polynomial and factoring it out.
  • Difference of Squares: This applies to polynomials in the form a² - b², which can be factored as (a + b)(a - b).
  • Perfect Square Trinomials: These are trinomials that can be factored into the form (a + b)² or (a - b)².
  • Factoring by Grouping: This technique is often used for polynomials with four or more terms and involves grouping terms together that have common factors.
  • Trial and Error: For quadratic trinomials (ax² + bx + c), trial and error can be used to find two binomials that multiply to give the trinomial.

In our case, we'll be using a combination of these techniques to factor the polynomial b² + 3b − b⁴ − 3b³ / b² − 9. Remember, practice makes perfect! The more you work with factoring polynomials, the more comfortable and confident you'll become.

2. Rewriting and Simplifying the Polynomial

The first step in factoring any polynomial is often to rewrite and simplify it. This makes the structure of the polynomial clearer and can help us identify potential factoring techniques. Our polynomial is b² + 3b − b⁴ − 3b³ / b² − 9. Notice something important here, guys! We have a division by b² − 9. To avoid confusion and make the factoring process smoother, we'll treat the entire expression as:

(b² + 3b − b⁴ − 3b³) / (b² − 9)

This is crucial! Now, let's focus on the numerator: b² + 3b − b⁴ − 3b³. To make it easier to work with, we'll rearrange the terms in descending order of their exponents:

−b⁴ − 3b³ + b² + 3b

This arrangement helps us identify potential patterns and apply factoring techniques more effectively. It’s like organizing your tools before starting a project – it makes the job much easier! Now, we're ready to move on to the next step: looking for common factors.

3. Factoring out the Greatest Common Factor (GCF)

The next step in our factoring journey is to look for the Greatest Common Factor (GCF) in the numerator. The GCF is the largest factor that divides all terms in the polynomial. In our rearranged numerator, −b⁴ − 3b³ + b² + 3b, we can see that each term has at least one 'b' in it. Also, notice the negative sign in the first term. It's often a good idea to factor out a negative sign if the leading coefficient (the coefficient of the highest-degree term) is negative.

So, let's factor out a -b from each term:

-b (b³ + 3b² - b - 3)

Factoring out the GCF has significantly simplified our expression. We've reduced the degree of the polynomial inside the parentheses, which makes it easier to work with. This is a classic factoring technique, and it's often the first thing you should look for when factoring any polynomial. It's like finding a common thread that runs through all the terms, allowing us to pull them apart and reorganize them. Now, we have a simpler polynomial inside the parentheses, but we're not done yet! We still need to factor b³ + 3b² - b - 3. This is where the next technique comes into play: factoring by grouping.

4. Factoring by Grouping

Now we're faced with factoring the polynomial b³ + 3b² - b - 3. Since it has four terms, factoring by grouping is a promising strategy. Factoring by grouping involves pairing terms together and factoring out a common factor from each pair. The goal is to create a common binomial factor that can then be factored out from the entire expression.

Let's group the first two terms and the last two terms:

(b³ + 3b²) + (-b - 3)

From the first group, (b³ + 3b²), we can factor out a b²:

b² (b + 3)

From the second group, (-b - 3), we can factor out a -1:

-1 (b + 3)

Now we have:

b²(b + 3) - 1(b + 3)

Notice that both terms now have a common binomial factor of (b + 3). This is exactly what we were hoping for! We can now factor out the (b + 3) from the entire expression:

(b + 3)(b² - 1)

Wow, guys, we've made significant progress! We've successfully factored the cubic polynomial into the product of a binomial and a quadratic. But we're not quite finished yet. Notice that the quadratic factor, (b² - 1), looks familiar. It's a special form that we can factor further.

5. Recognizing and Applying the Difference of Squares

Our expression now looks like this: -b(b + 3)(b² - 1). Take a closer look at the factor (b² - 1). This is a classic example of the difference of squares pattern. The difference of squares pattern states that for any two terms a and b:

a² - b² = (a + b)(a - b)

In our case, b² - 1 can be seen as b² - 1², where a = b and b = 1. So, we can apply the difference of squares pattern:

b² - 1 = (b + 1)(b - 1)

This is a powerful factoring shortcut, and it's essential to be able to recognize it quickly. It's like having a key that unlocks a door – once you see the pattern, the factoring becomes much simpler. Now, let's substitute this back into our expression:

-b(b + 3)(b + 1)(b - 1)

We've completely factored the numerator of our original expression! This is a major accomplishment. But remember, we still have the denominator to consider. Let's bring it back into the picture and see how it fits in.

6. Factoring the Denominator

Let's not forget about our denominator, which was b² − 9. This expression also fits the difference of squares pattern! We can rewrite it as b² - 3², where a = b and b = 3. Applying the difference of squares pattern, we get:

b² - 9 = (b + 3)(b - 3)

Now we have factored both the numerator and the denominator of our original expression. This is like having all the pieces of a puzzle – now we can put them together and see the final picture.

7. Putting It All Together and Simplifying

Now we have the fully factored numerator and denominator. Let's write out the entire expression:

(-b(b + 3)(b + 1)(b - 1)) / ((b + 3)(b - 3))

This is where the magic happens! Notice that we have a common factor of (b + 3) in both the numerator and the denominator. This means we can cancel them out!

(-b(b + 1)(b - 1)) / (b - 3)

Canceling common factors is a crucial step in simplifying rational expressions (expressions that are fractions with polynomials). It's like removing the unnecessary pieces that are cluttering the expression and revealing its simplest form. We're almost there, guys! We've simplified the expression significantly, but let's see if we can go any further.

8. Final Simplified Form

Our expression now looks like (-b(b + 1)(b - 1)) / (b - 3). Let's expand the numerator to see if we can simplify it further. First, we'll multiply (b + 1)(b - 1). This is another difference of squares pattern, but in reverse:

(b + 1)(b - 1) = b² - 1

Now, we'll multiply this by -b:

-b(b² - 1) = -b³ + b

So, our final simplified expression is:

(-b³ + b) / (b - 3)

We've reached the end of our factoring journey! We started with a complex polynomial and, through a series of systematic steps, simplified it to its most basic form. This final expression represents the factored form of the original polynomial. You guys did an amazing job following along! Remember, factoring is a skill that improves with practice. The more you work with different types of polynomials, the more comfortable and confident you'll become. Keep practicing, and you'll be a factoring pro in no time!

Conclusion

Factoring the polynomial b² + 3b − b⁴ − 3b³ / b² − 9 involves a series of steps, including rewriting the expression, factoring out the GCF, factoring by grouping, recognizing the difference of squares pattern, and simplifying the resulting expression. By systematically applying these techniques, we successfully factored the polynomial and arrived at the simplified form (-b³ + b) / (b - 3). This process highlights the importance of understanding different factoring techniques and knowing when to apply them. With practice, you can master the art of polynomial factoring and confidently tackle complex algebraic problems. Keep up the great work!