(x+1)^4 Expand Formula

Jasonbradley

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Jun 16, 2024

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Expanding (x+1)^4

The expansion of (x+1)^4 can be done using the binomial theorem or by repeated multiplication.

Expanding using the Binomial Theorem

The binomial theorem states that:

(x + y)^n = ∑ (n choose k) * x^(n-k) * y^k

where:

• n is a non-negative integer (the power of the binomial)
• k is an integer that ranges from 0 to n
• (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)

Applying this to (x+1)^4:

1. n = 4
2. k goes from 0 to 4

Therefore, the expansion is:

(x + 1)^4 = (4 choose 0) * x^4 * 1^0 + (4 choose 1) * x^3 * 1^1 + (4 choose 2) * x^2 * 1^2 + (4 choose 3) * x^1 * 1^3 + (4 choose 4) * x^0 * 1^4

Calculating the binomial coefficients:

• (4 choose 0) = 1
• (4 choose 1) = 4
• (4 choose 2) = 6
• (4 choose 3) = 4
• (4 choose 4) = 1

Substituting the coefficients:

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

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

Expanding by Repeated Multiplication

We can also expand (x+1)^4 by repeatedly multiplying:

(x + 1)^4 = (x + 1) * (x + 1) * (x + 1) * (x + 1)

1. Multiply the first two factors:

   (x + 1) * (x + 1) = x^2 + 2x + 1

2. Multiply the result by the third factor:

   (x^2 + 2x + 1) * (x + 1) = x^3 + 3x^2 + 3x + 1

3. Multiply the result by the fourth factor:

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

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

Conclusion

Both methods lead to the same result: (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1. The binomial theorem is a more efficient method for higher powers, while repeated multiplication is simpler for smaller powers.