Math Problem Statement

In the expansion of (2x - 1)^30, find the coefficients of x^2 and x^3.

Solution

The given image shows a question asking for the coefficients of x2x^2 and x3x^3 in the expansion of (2x1)30(2x - 1)^{30}.

We will use the Binomial Theorem to expand this expression. The Binomial Theorem states:

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

For the given expression (2x1)30(2x - 1)^{30}:

  • a=2xa = 2x
  • b=1b = -1
  • n=30n = 30

The general term in the expansion is:

Tk=(30k)(2x)30k(1)kT_k = \binom{30}{k} (2x)^{30-k} (-1)^k

This term simplifies to:

Tk=(30k)230kx30k(1)kT_k = \binom{30}{k} 2^{30-k} x^{30-k} (-1)^k

Finding the coefficient of x2x^2:

We need to find the term where the power of xx is 2. That means 30k=230 - k = 2, so k=28k = 28. Substituting k=28k = 28 into the general term:

T28=(3028)22x2(1)28T_{28} = \binom{30}{28} 2^2 x^2 (-1)^{28}

Simplifying:

T28=(3028)4x2T_{28} = \binom{30}{28} 4 x^2

Since (3028)=(302)=30×292=435\binom{30}{28} = \binom{30}{2} = \frac{30 \times 29}{2} = 435, the coefficient of x2x^2 is:

4×435=17404 \times 435 = 1740

Finding the coefficient of x3x^3:

We need to find the term where the power of xx is 3. That means 30k=330 - k = 3, so k=27k = 27. Substituting k=27k = 27 into the general term:

T27=(3027)23x3(1)27T_{27} = \binom{30}{27} 2^3 x^3 (-1)^{27}

Simplifying:

T27=(3027)8x3(1)T_{27} = \binom{30}{27} 8 x^3 (-1)

Since (3027)=(303)=30×29×286=4060\binom{30}{27} = \binom{30}{3} = \frac{30 \times 29 \times 28}{6} = 4060, the coefficient of x3x^3 is:

8×4060=32,480-8 \times 4060 = -32,480

Final Answers:

  • The coefficient of x2x^2 is 1740.
  • The coefficient of x3x^3 is -32,480.

Would you like further details or have any questions about this expansion?

Here are five related questions to explore:

  1. How do binomial coefficients change when the exponent increases?
  2. Can you find the coefficient of x4x^4 in the expansion of (2x1)30(2x - 1)^{30}?
  3. What is the sum of all the coefficients in the expansion of (2x1)30(2x - 1)^{30}?
  4. How would the expansion differ if the expression were (2x+1)30(2x + 1)^{30} instead?
  5. What is the general form of the binomial theorem for negative exponents?

Tip: The binomial coefficients can be quickly calculated using Pascal's triangle or the formula for combinations (nk)=n!k!(nk)!\binom{n}{k} = \frac{n!}{k!(n-k)!}.

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Math Problem Analysis

Mathematical Concepts

Binomial Expansion
Polynomial Coefficients
Combinatorics

Formulas

Binomial Expansion: (a + b)^n = ∑(n choose k) * a^(n-k) * b^k
Combination Formula: (n choose k) = n! / [k!(n - k)!]

Theorems

Binomial Theorem

Suitable Grade Level

Grades 11-12