(x+4)^5 Expand

Jasonbradley

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Jun 16, 2024

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Expanding (x + 4)^5

The expansion of (x + 4)^5 can be achieved using the Binomial Theorem, a powerful tool for expanding expressions of the form (a + b)^n.

The Binomial Theorem

The Binomial Theorem states that for any real numbers a and b, and any non-negative integer n:

(a + b)^n = n C 0 a^n b^0 + n C 1 a^(n-1) b^1 + n C 2 a^(n-2) b^2 + ... + n C (n-1) a^1 b^(n-1) + n C n a^0 b^n

Where n C k represents the binomial coefficient, calculated as:

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

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

1. Identify a and b: In our case, a = x and b = 4.

2. Identify n: n = 5.

3. Calculate the binomial coefficients:

   • 5 C 0 = 5! / (0! * 5!) = 1
   • 5 C 1 = 5! / (1! * 4!) = 5
   • 5 C 2 = 5! / (2! * 3!) = 10
   • 5 C 3 = 5! / (3! * 2!) = 10
   • 5 C 4 = 5! / (4! * 1!) = 5
   • 5 C 5 = 5! / (5! * 0!) = 1

4. Apply the formula: (x + 4)^5 = 1 * x^5 * 4^0 + 5 * x^4 * 4^1 + 10 * x^3 * 4^2 + 10 * x^2 * 4^3 + 5 * x^1 * 4^4 + 1 * x^0 * 4^5

5. Simplify: (x + 4)^5 = x^5 + 20x^4 + 160x^3 + 640x^2 + 1280x + 1024

Therefore, the expanded form of (x + 4)^5 is: x^5 + 20x^4 + 160x^3 + 640x^2 + 1280x + 1024