Binomial Expansion: What's The Expression?

Hey guys! Today, we're diving into the fascinating world of binomial expansions. We have a bit of a puzzle on our hands: figuring out which binomial expression expands to give us $243 c^5+810 c^4 d+1080 c^3 d^2+720 c^2 d^3+240 c d^4+32 d^5$. It might look intimidating at first, but don't worry, we'll break it down step by step and make it super clear. Think of it like this: we're reverse-engineering the expansion, which can be a fun and rewarding challenge! So, let's put on our math hats and get started on this journey of discovery!

Understanding Binomial Expansion

Before we jump into solving the specific problem, let's do a quick refresh on binomial expansion. Binomial expansion is essentially the process of expanding an expression in the form of $(a + b)^n$, where 'a' and 'b' are terms, and 'n' is a positive integer. The binomial theorem provides a formula for this expansion, making it systematic and predictable. The general formula for the binomial theorem is:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Where $\binom{n}{k}$ represents the binomial coefficient, often read as "n choose k," and it's calculated as:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Here, '!' denotes the factorial, which means the product of all positive integers up to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1). So, when we expand a binomial like $(a + b)^n$, we get a series of terms. Each term consists of a binomial coefficient, a power of 'a', and a power of 'b'. The powers of 'a' decrease from n to 0, while the powers of 'b' increase from 0 to n. The binomial coefficients give us the numerical factors in each term. This understanding is crucial because it gives us a framework to work with. Recognizing patterns, like how the exponents change and how the binomial coefficients arise, will help us immensely in identifying the original binomial expression.

Key Patterns in Binomial Expansion

To successfully identify the binomial expression, it's important to understand the key patterns that emerge during expansion. Let's explore those patterns:

1. Number of Terms: When you expand $(a + b)^n$, you'll always end up with n + 1 terms. For example, $(a + b)^5$ will have 6 terms. This gives us a quick way to check if our expanded expression has the correct number of terms.
2. Exponents: The exponents of 'a' start at n and decrease by 1 with each term, while the exponents of 'b' start at 0 and increase by 1 with each term. This creates a predictable progression that helps us match terms during reverse engineering.
3. Binomial Coefficients: These coefficients follow a symmetrical pattern. They start small, increase to a maximum in the middle, and then decrease again. These coefficients can be easily determined using Pascal's Triangle, where each number is the sum of the two numbers directly above it. For example, the coefficients for the expansion of a power of 5 (like we have in our question) are 1, 5, 10, 10, 5, and 1. Recognizing this symmetry can save us a lot of calculation time.
4. First and Last Terms: The first term will always be $a^n$, and the last term will always be $b^n$. This gives us direct clues about the 'a' and 'b' terms in the binomial.

By keeping these patterns in mind, we can dissect the expanded expression more efficiently and narrow down the possible binomial candidates.

Analyzing the Given Expression

Okay, let's get back to our specific problem. We have the expanded expression:

$243 c^5+810 c^4 d+1080 c^3 d^2+720 c^2 d^3+240 c d^4+32 d^5$

Our mission is to figure out which binomial, when raised to the power of 5, gives us this expression. To do this, we'll use the patterns we discussed earlier. First, notice that there are 6 terms in the expression. This tells us that we are dealing with an expansion of the form $(a + b)^5$, which confirms the power to which the binomial is raised.

Identifying 'a' and 'b'

Now, let's identify the 'a' and 'b' terms. Remember, the first term in the expansion corresponds to $a^5$, and the last term corresponds to $b^5$. In our expression:

• The first term is $243c^5$. We need to find the fifth root of 243, which is 3. So, 'a' likely involves $3c$.
• The last term is $32d^5$. The fifth root of 32 is 2. So, 'b' likely involves $2d$.

This gives us a strong clue that our binomial might look something like $(3c + 2d)$ or $(3c - 2d)$. We'll need to be careful about the signs, but for now, this is a great starting point. The coefficients also play a significant role. The binomial coefficients for an expansion to the power of 5 are 1, 5, 10, 10, 5, and 1. These come from Pascal's Triangle, and we should see these (or multiples of them) in our expansion.

Verifying the Coefficients

Let's check the coefficients in our expression against the binomial coefficients. The coefficients in our expanded expression are 243, 810, 1080, 720, 240, and 32. We need to see if these coefficients align with the binomial coefficients (1, 5, 10, 10, 5, 1) for $n = 5$.

• The first term's coefficient is 243, which is $3^5$, matching our 'a' term.
• The last term's coefficient is 32, which is $2^5$, matching our 'b' term.
• Now, let’s look at the second term: 810. If our binomial is $(3c + 2d)^5$, the second term should be $5 * (3c)^4 * (2d)$. Let's calculate this:
  • $5 * (81c^4) * (2d) = 810c^4d$. This matches perfectly!
• The third term should be $10 * (3c)^3 * (2d)^2$. Let’s calculate:
  • $10 * (27c^3) * (4d^2) = 1080c^3d^2$. Again, a perfect match!

It looks like we're on the right track. By calculating a few terms and comparing them with the coefficients and exponents in the expanded expression, we can build confidence in our identified binomial. Now, let’s move forward with this insight.

Identifying the Correct Binomial

Based on our analysis, we've narrowed down the possibilities. We strongly suspect that the binomial is $(3c + 2d)^5$. We've already verified the first few terms, but let's solidify our conclusion by examining the remaining terms and making sure everything lines up perfectly.

Continuing the Verification

We’ve checked the first three terms, so let's move on to the fourth term in the expansion, which should be:

$\binom{5}{3}(3c)^2(2d)^3 = 10 * (9c^2) * (8d^3) = 720c^2d^3$

Looking back at our given expression, the fourth term is indeed $720c^2d^3$. This further strengthens our belief that we've correctly identified the binomial.

Now, let’s check the fifth term:

$\binom{5}{4}(3c)^1(2d)^4 = 5 * (3c) * (16d^4) = 240cd^4$

This matches the fifth term in our expression as well! And we already know the last term, $32d^5$, matches.

Final Confirmation

So, we have meticulously checked each term, and they all align perfectly with the expansion of $(3c + 2d)^5$. This detailed verification leaves us with a high degree of confidence in our answer. We've successfully reverse-engineered the binomial expansion!

Conclusion

After a thorough analysis and step-by-step verification, we've confidently determined that the expression $243 c^5+810 c^4 d+1080 c^3 d^2+720 c^2 d^3+240 c d^4+32 d^5$ is the expansion of the binomial $(3c + 2d)^5$. Understanding the underlying principles of binomial expansion, recognizing patterns, and meticulously verifying each term were key to solving this problem.

Remember, guys, breaking down complex problems into smaller, manageable steps is a powerful strategy. Keep practicing, and you'll become binomial expansion masters in no time! Math can be a rewarding journey of discovery, and problems like these are great opportunities to sharpen your skills and deepen your understanding. So, keep exploring, keep learning, and most importantly, keep having fun with math!