Finding The Coefficient Of X⁴ In (x+3)¹² A Step-by-Step Guide

by Chloe Fitzgerald 62 views

Hey guys! Today, we're diving into the fascinating world of binomial expansions to uncover the coefficient of the x⁴ term in the expansion of (x+3)¹². This might sound like a mouthful, but don't worry, we'll break it down step-by-step so you can master this concept. Understanding binomial expansions is crucial in various areas of mathematics, from algebra to calculus, and even in probability and statistics. So, let's embark on this mathematical journey together!

Understanding Binomial Expansion

At the heart of our quest lies the binomial theorem, a powerful tool that allows us to expand expressions of the form (a+b)ⁿ, where n is a non-negative integer. The binomial theorem provides a systematic way to determine the coefficients and terms in the expansion. The general formula for the binomial theorem is:

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

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

(nk)=n!k!(nk)!{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 instance, 5! = 5 × 4 × 3 × 2 × 1 = 120. The binomial coefficient ${n \choose k}$ tells us the number of ways to choose k objects from a set of n distinct objects. This concept is fundamental in combinatorics, a branch of mathematics dealing with counting and arrangements.

To truly grasp the binomial theorem, let's consider a simple example. Expanding (x + y)² using the binomial theorem, we have:

(x+y)2=(20)x2y0+(21)x1y1+(22)x0y2(x + y)^2 = {2 \choose 0}x^2y^0 + {2 \choose 1}x^1y^1 + {2 \choose 2}x^0y^2

Calculating the binomial coefficients:

  • (20)=2!0!2!=1{2 \choose 0} = \frac{2!}{0!2!} = 1

  • (21)=2!1!1!=2{2 \choose 1} = \frac{2!}{1!1!} = 2

  • (22)=2!2!0!=1{2 \choose 2} = \frac{2!}{2!0!} = 1

Substituting these values back into the expansion, we get:

(x+y)2=1x2+2xy+1y2=x2+2xy+y2(x + y)^2 = 1x^2 + 2xy + 1y^2 = x^2 + 2xy + y^2

This familiar result demonstrates the power and elegance of the binomial theorem. Now, let's apply this knowledge to our original problem: finding the coefficient of the x⁴ term in the expansion of (x+3)¹².

Applying the Binomial Theorem to (x+3)¹²

Now, let's tackle the main problem. We want to find the coefficient of the x⁴ term in the binomial expansion of (x+3)¹². Using the binomial theorem, we know that the general term in the expansion of (x+3)¹² is given by:

(12k)x12k3k{12 \choose k} x^{12-k} 3^k

Where k ranges from 0 to 12. Our goal is to find the value of k that results in an x⁴ term. This means we need to solve the equation:

12k=412 - k = 4

Solving for k, we get:

k=124=8k = 12 - 4 = 8

So, the term we're interested in is the one where k = 8. Plugging this value into the general term formula, we get:

(128)x12838=(128)x438{12 \choose 8} x^{12-8} 3^8 = {12 \choose 8} x^4 3^8

Therefore, the coefficient of the x⁴ term is ${12 \choose 8} 3^8$. This matches option D in the given choices. But just to be super clear, let's calculate this coefficient. First, let's compute the binomial coefficient:

(128)=12!8!(128)!=12!8!4!=12×11×10×94×3×2×1=495{12 \choose 8} = \frac{12!}{8!(12-8)!} = \frac{12!}{8!4!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495

Next, we calculate 3⁸:

38=65613^8 = 6561

Finally, we multiply these two values together:

495×6561=3,247,695495 \times 6561 = 3,247,695

So, the coefficient of the x⁴ term in the expansion of (x+3)¹² is 3,247,695. This confirms that option D, ${12 \choose 8}(3)^8$, is indeed the correct answer. Remember, the binomial theorem is a powerful tool that simplifies the expansion of binomial expressions, saving us from tedious manual multiplication.

Why Other Options are Incorrect

Now that we've confidently identified the correct answer, let's take a moment to understand why the other options are incorrect. This will solidify our understanding of the binomial theorem and prevent similar errors in the future.

  • Option A: ${12 \choose 7}(3)^7$

    This option represents the coefficient of the term where x has a power of 5. To see this, let's plug k = 7 into the general term formula:

    (127)x12737=(127)x537{12 \choose 7} x^{12-7} 3^7 = {12 \choose 7} x^5 3^7

    Since we're looking for the x⁴ term, this option is incorrect.

  • Option B: ${12 \choose 8}$

    This option only includes the binomial coefficient but omits the crucial factor of 3 raised to the power of 8. Remember, in the binomial expansion of (x+3)¹², the second term (3 in this case) also contributes to the coefficient. Therefore, this option is incomplete and incorrect.

  • Option C: ${12 \choose 4}(3)^4$

    This option seems to confuse the power of x with the power of 3. While ${12 \choose 4}$ is a valid binomial coefficient in the expansion, the exponent of 3 should correspond to the value of k that gives us the x⁴ term, which we already determined to be 8. If we calculate the term corresponding to k=4 we get

    (124)x12434=(124)x834{12 \choose 4} x^{12-4} 3^4 = {12 \choose 4} x^8 3^4

    Thus, this option represents the coefficient of the x⁸ term, not the x⁴ term. So, by carefully analyzing each option and understanding the role of both the binomial coefficient and the constant term in the expansion, we can confidently eliminate the incorrect choices.

Key Takeaways and Further Practice

Alright, guys, we've successfully navigated through the binomial expansion and found the coefficient of the x⁴ term in (x+3)¹². Let's recap the key takeaways from this problem:

  1. The Binomial Theorem is Your Friend: The binomial theorem provides a systematic way to expand expressions of the form (a+b)ⁿ. Remember the general formula and how to calculate binomial coefficients.
  2. Identify the Correct Term: To find the coefficient of a specific term (like x⁴), determine the value of k that satisfies the condition for the exponent of x.
  3. Don't Forget the Constant Term: In binomial expansions like (x+3)¹², the constant term (3 in this case) also contributes to the coefficient. Make sure to include it in your calculations.
  4. Practice Makes Perfect: The more you practice binomial expansion problems, the more comfortable and confident you'll become. Try different values of n and different constants to challenge yourself.

To further solidify your understanding, here are a few practice problems:

  1. Find the coefficient of the x³ term in the expansion of (x+2)⁷.
  2. What is the constant term in the expansion of (2x - 1/x)¹⁰?
  3. Determine the coefficient of the a⁵b³ term in the expansion of (a - 2b)⁸.

By working through these problems, you'll reinforce your knowledge and develop a deeper understanding of the binomial theorem. Remember, mathematics is a journey of exploration and discovery. Keep practicing, keep asking questions, and keep having fun!

In conclusion, we've successfully found that the coefficient of the x⁴ term in the binomial expansion of (x+3)¹² is ${12 \choose 8}(3)^8$. By understanding the binomial theorem and carefully applying it, we can confidently solve these types of problems. So, keep exploring the fascinating world of mathematics, and I'll catch you in the next problem-solving adventure! Happy learning!