Math Problem Statement

Expand the binomial (x+2)10

Solution

To expand the binomial (x+2)10(x + 2)^{10}, we use the Binomial Theorem. The Binomial Theorem states:

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

For (x+2)10(x + 2)^{10}:

  • a=xa = x
  • b=2b = 2
  • n=10n = 10

Thus, the expansion is:

(x+2)10=k=010(10k)x10k2k(x + 2)^{10} = \sum_{k=0}^{10} \binom{10}{k} \cdot x^{10-k} \cdot 2^k

I'll calculate the expansion step by step.The expanded form of (x+2)10(x + 2)^{10} is:

x10+20x9+180x8+960x7+3360x6+8064x5+13440x4+15360x3+11520x2+5120x+1024x^{10} + 20x^9 + 180x^8 + 960x^7 + 3360x^6 + 8064x^5 + 13440x^4 + 15360x^3 + 11520x^2 + 5120x + 1024

Would you like a detailed explanation of this expansion or have any other questions? Here are five related questions you might find useful:

  1. How can you apply the Binomial Theorem to negative exponents?
  2. What are the coefficients in the expansion of (x3)8(x - 3)^8?
  3. How is the Binomial Theorem related to Pascal's Triangle?
  4. Can you find the middle term in the expansion of (x+1)7(x + 1)^7?
  5. How would the expansion change if the binomial were (2x+3)10(2x + 3)^{10}?

Tip: The coefficients of each term in a binomial expansion correspond to the entries in Pascal's Triangle for that power.

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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} \cdot a^{n-k} \cdot b^k

Theorems

Binomial Theorem

Suitable Grade Level

Grades 11-12