Math Problem Statement

Solve (2x+1)^4

Solution

To solve the expression (2x+1)4(2x + 1)^4, we will expand it using the Binomial Theorem.

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer nn:

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

In this case:

  • a=2xa = 2x
  • b=1b = 1
  • n=4n = 4

Step 2: Apply the Binomial Theorem

Expanding (2x+1)4(2x + 1)^4:

(2x+1)4=k=04(4k)(2x)4k(1)k(2x + 1)^4 = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} (1)^k

Let's calculate each term in the expansion:

  • For k=0k = 0: (40)(2x)4(1)0=116x41=16x4\binom{4}{0} (2x)^4 (1)^0 = 1 \cdot 16x^4 \cdot 1 = 16x^4

  • For k=1k = 1: (41)(2x)3(1)1=48x31=32x3\binom{4}{1} (2x)^3 (1)^1 = 4 \cdot 8x^3 \cdot 1 = 32x^3

  • For k=2k = 2: (42)(2x)2(1)2=64x21=24x2\binom{4}{2} (2x)^2 (1)^2 = 6 \cdot 4x^2 \cdot 1 = 24x^2

  • For k=3k = 3: (43)(2x)1(1)3=42x1=8x\binom{4}{3} (2x)^1 (1)^3 = 4 \cdot 2x \cdot 1 = 8x

  • For k=4k = 4: (44)(2x)0(1)4=111=1\binom{4}{4} (2x)^0 (1)^4 = 1 \cdot 1 \cdot 1 = 1

Step 3: Combine all the terms

The expansion is:

(2x+1)4=16x4+32x3+24x2+8x+1(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1

Conclusion

So, the expanded form of (2x+1)4(2x + 1)^4 is:

16x4+32x3+24x2+8x+1\boxed{16x^4 + 32x^3 + 24x^2 + 8x + 1}

Would you like more details on the binomial expansion? Here are some related questions you can try:

  1. How would you expand (3x2)3(3x - 2)^3 using the Binomial Theorem?
  2. What is the general term in the expansion of (a+b)n(a + b)^n?
  3. Can you expand (x+1/x)4(x + 1/x)^4 using the same method?
  4. How do the coefficients in the binomial expansion relate to Pascal's Triangle?
  5. What happens when xx is a complex number in the binomial expansion?

Tip: When expanding binomials, make sure to correctly calculate each binomial coefficient using (nk)\binom{n}{k} and carefully apply the powers to each term.

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Math Problem Analysis

Mathematical Concepts

Binomial Theorem
Binomial Expansion

Formulas

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

Theorems

Binomial Theorem

Suitable Grade Level

Grades 10-12