Finding The Fifth Term In The Binomial Expansion Of (x+5)^8

Hey there, math enthusiasts! Ever found yourself staring at a binomial expansion problem, feeling like you're decoding an ancient script? Well, fear not! Today, we're going to crack the code and figure out the fifth term in the binomial expansion of $(x+5)^8$. It might sound intimidating, but trust me, we'll break it down step by step, making it as clear as a sunny day. So, grab your thinking caps, and let's dive in!

Understanding the Binomial Theorem

Before we jump into the problem, let's quickly revisit the binomial theorem. This theorem is our trusty map when we're expanding expressions like $(a + b)^n$. In essence, it provides a formula to find any specific term in the expansion without having to multiply the entire expression out. Think of it as a shortcut through the expansion jungle! The binomial theorem is given by:

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

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

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

Here, "!" denotes the factorial, where $n!$ (n factorial) is the product of all positive integers up to $n$. For example, $5! = 5 × 4 × 3 × 2 × 1 = 120$. The binomial coefficient tells us the numerical coefficient of each term in the expansion. Understanding this formula is crucial as it forms the backbone of solving binomial expansion problems. It might look a bit complex at first, but with a little practice, it'll become second nature.

Now, let's break down the components of this formula and see how they apply to our specific problem. We have $(a + b)^n$, where:

• a is the first term in the binomial
• b is the second term in the binomial
• n is the power to which the binomial is raised

In our case, we have $(x + 5)^8$, so a = x, b = 5, and n = 8. The k in the formula represents the term number we're looking for, starting from 0. So, if we want the first term, k = 0; for the second term, k = 1; and so on. This is a key point to remember, as it will help us pinpoint the correct values when we calculate the fifth term. The beauty of the binomial theorem lies in its ability to isolate any specific term without having to expand the entire expression, which can be a real time-saver!

Applying the Theorem to Find the Fifth Term

Okay, now that we've got the binomial theorem under our belts, let's apply it to our problem. We're looking for the fifth term in the expansion of $(x + 5)^8$. Remember, since we start counting terms from 0, the fifth term corresponds to $k = 4$. Think of it like this: the first term is k = 0, the second is k = 1, the third is k = 2, the fourth is k = 3, and finally, the fifth term is k = 4. It's a common mistake to think the fifth term means k = 5, so always double-check!

Now, let's plug the values into the binomial theorem formula. We have:

• a = x
• b = 5
• n = 8
• k = 4

So, the term we're looking for is:

${8 \choose 4} x^{8-4} 5^4$

First, let's calculate the binomial coefficient ${8 \choose 4}$:

${8 \choose 4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{(4 × 3 × 2 × 1)(4 × 3 × 2 × 1)}$

We can simplify this by canceling out the common factors:

${8 \choose 4} = \frac{8 × 7 × 6 × 5}{4 × 3 × 2 × 1} = 70$

Great! Now we know that the binomial coefficient for the fifth term is 70. Next, let's calculate the powers of x and 5:

• $x^{8-4} = x^4$
• $5^4 = 5 × 5 × 5 × 5 = 625$

Now we have all the pieces of the puzzle. Let's put them together:

Fifth term = ${8 \choose 4} x^{8-4} 5^4 = 70 × x^4 × 625$

Multiply the numbers together:

Fifth term = $70 × 625 × x^4 = 43,750 x^4$

So, the fifth term in the binomial expansion of $(x + 5)^8$ is $43,750x^4$. High five! We've successfully navigated the binomial theorem and found our answer.

Comparing with the Given Options

Alright, we've calculated the fifth term to be $43,750x^4$. Now, let's compare this with the options given in the problem:

A. $175,000 x^3$ B. $43,750 x^4$ C. $3,125 x^5$ D. $7,000 x^5$

Looking at the options, we can see that option B, $43,750 x^4$, matches our calculated result perfectly. Therefore, the correct answer is B. It's always a good idea to double-check your answer, especially in math problems. Make sure the coefficient and the power of the variable match your calculations.

Common Mistakes to Avoid

Before we wrap up, let's quickly touch on some common mistakes students make when working with binomial expansions. Being aware of these pitfalls can save you from making errors in the future.

1. Incorrectly Identifying the Term Number: As we discussed earlier, the term number starts from 0, not 1. So, the fifth term corresponds to $k = 4$, not $k = 5$. This is a very common mistake, so always double-check which term you're looking for.
2. Miscalculating the Binomial Coefficient: The binomial coefficient formula can look intimidating, and it's easy to make mistakes in the factorial calculations. Take your time, write out the factorials, and cancel out common factors carefully. It's also helpful to use a calculator for larger factorials to avoid errors.
3. Forgetting to Raise Both Terms to the Correct Powers: In the binomial theorem formula, both a and b need to be raised to the correct powers. Make sure you're using n - k for the power of a and k for the power of b. A simple oversight here can lead to a completely wrong answer.
4. Arithmetic Errors: Basic arithmetic errors, like multiplying or adding numbers incorrectly, can also lead to wrong answers. It's a good practice to double-check your calculations, especially in the final steps, to ensure accuracy. Pay close attention to detail, and don't rush through the calculations.

Real-World Applications of Binomial Expansion

Now, you might be wondering,