Math Problem Statement

Find the 4th term in the expansion of (a+b)^3 In simplest form

Solution

The expression given is (a+b)3(a + b)^3, and we need to find the 4th term in its expansion.

The binomial expansion of (a+b)n(a + b)^n is given by:

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

For (a+b)3(a + b)^3, the expansion will be:

(a+b)3=(30)a3b0+(31)a2b1+(32)a1b2+(33)a0b3(a + b)^3 = \binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3}a^0b^3

Simplifying, we get:

(a+b)3=1a3+3a2b+3ab2+1b3(a + b)^3 = 1 \cdot a^3 + 3 \cdot a^2b + 3 \cdot ab^2 + 1 \cdot b^3

So, the terms of the expansion are:

  1. a3a^3
  2. 3a2b3a^2b
  3. 3ab23ab^2
  4. b3b^3

Therefore, the 4th term in the expansion is b3\mathbf{b^3}.

Would you like details on binomial coefficients, or have any questions?

Here are 5 related questions:

  1. What is the 3rd term in the expansion of (ab)4(a - b)^4?
  2. How do you calculate the binomial coefficient (nk)\binom{n}{k}?
  3. What is the general term formula in the binomial expansion of (a+b)n(a + b)^n?
  4. How can the binomial theorem be applied to solve problems in probability?
  5. How would the expansion differ if the expression were (a+b)5(a + b)^5?

Tip: In binomial expansions, each term is structured as a product of a binomial coefficient, one variable raised to a power, and the other variable raised to the complementary power.

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Math Problem Analysis

Mathematical Concepts

Binomial Theorem
Binomial Coefficients

Formulas

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

Theorems

Binomial Theorem

Suitable Grade Level

High School