(2x+5)^4

Jasonbradley

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Jun 16, 2024

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Expanding (2x + 5)^4: A Step-by-Step Guide

Expanding expressions like (2x + 5)^4 can seem daunting at first, but with the right method, it becomes manageable. Here's a breakdown of how to do it using the binomial theorem:

Understanding the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n:

(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n

where (n choose k) represents the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Applying the Theorem to (2x + 5)^4

1. Identify a and b: In our case, a = 2x and b = 5.

2. Determine n: n = 4.

3. Calculate the Binomial Coefficients:

   • (4 choose 0) = 4! / (0! * 4!) = 1
   • (4 choose 1) = 4! / (1! * 3!) = 4
   • (4 choose 2) = 4! / (2! * 2!) = 6
   • (4 choose 3) = 4! / (3! * 1!) = 4
   • (4 choose 4) = 4! / (4! * 0!) = 1

4. Apply the Formula:

(2x + 5)^4 = (2x)^4 + (4 choose 1)(2x)^3(5) + (4 choose 2)(2x)^2(5)^2 + (4 choose 3)(2x)(5)^3 + (5)^4

5. Simplify:

(2x + 5)^4 = 16x^4 + 160x^3 + 600x^2 + 1000x + 625

Key Points

• Understanding the Binomial Theorem: This is the foundation for expanding binomials.
• Calculating Binomial Coefficients: Using the formula is crucial for accurate expansion.
• Systematic Approach: Following a step-by-step method helps avoid errors.

By applying the binomial theorem and carefully calculating each term, you can successfully expand (2x + 5)^4 and other similar expressions.