(x+1)^4 Expanded

Jasonbradley

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Jun 16, 2024

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Expanding (x+1)^4

The expression (x+1)^4 represents the fourth power of the binomial (x+1). Expanding this expression means writing it out in a simplified form without exponents. We can use the Binomial Theorem to do this.

The Binomial Theorem

The Binomial Theorem provides a formula for expanding any binomial raised to a power:

(a + b)^n = a^n + nC1 a^(n-1)b + nC2 a^(n-2)b^2 + ... + nCn-1 ab^(n-1) + b^n

Where:

• n is the power to which the binomial is raised
• nC1, nC2, ..., nCn are the binomial coefficients, which can be calculated using the formula: nCk = n! / (k! * (n-k)!)

Expanding (x+1)^4

Let's apply the Binomial Theorem to expand (x+1)^4:

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

2. Identify n: The power is n = 4.

3. Calculate binomial coefficients: We need to calculate the following coefficients:

   • 4C0 = 4! / (0! * 4!) = 1
   • 4C1 = 4! / (1! * 3!) = 4
   • 4C2 = 4! / (2! * 2!) = 6
   • 4C3 = 4! / (3! * 1!) = 4
   • 4C4 = 4! / (4! * 0!) = 1

4. Substitute the values into the formula:

(x + 1)^4 = x^4 + 4C1 x^3 (1) + 4C2 x^2 (1)^2 + 4C3 x (1)^3 + (1)^4

5. Simplify:

(x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1

Therefore, the expanded form of (x+1)^4 is x^4 + 4x^3 + 6x^2 + 4x + 1.