(x+5)^4

Jasonbradley

4 min read

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Jun 17, 2024

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

Expanding expressions like (x+5)^4 can seem daunting at first, but there's a methodical way to approach it. We'll explore two common methods: the Binomial Theorem and repeated multiplication.

1. Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (x+y)^n. The formula is as follows:

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

where:

• (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This determines the coefficient of each term in the expansion.
• k ranges from 0 to n.

Let's apply this to our example (x+5)^4:

1. Determine n: In this case, n = 4.
2. Expand the summation: We will have terms where k = 0, 1, 2, 3, and 4.
3. Calculate 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. Substitute into the formula:
   • (x + 5)^4 = (1 * x^4 * 5^0) + (4 * x^3 * 5^1) + (6 * x^2 * 5^2) + (4 * x^1 * 5^3) + (1 * x^0 * 5^4)
5. Simplify:
   • (x + 5)^4 = x^4 + 20x^3 + 150x^2 + 500x + 625

2. Repeated Multiplication

This method involves expanding the expression step-by-step.

1. Start with (x + 5)^2:
   • (x + 5)^2 = (x + 5)(x + 5) = x^2 + 10x + 25
2. Multiply by (x + 5) again:
   • (x + 5)^3 = (x^2 + 10x + 25)(x + 5) = x^3 + 15x^2 + 75x + 125
3. Multiply by (x + 5) one more time:
   • (x + 5)^4 = (x^3 + 15x^2 + 75x + 125)(x + 5) = x^4 + 20x^3 + 150x^2 + 500x + 625

Conclusion

Both methods lead to the same result:

(x + 5)^4 = x^4 + 20x^3 + 150x^2 + 500x + 625

While the Binomial Theorem might seem more complex at first, it becomes particularly useful when dealing with higher powers of (x+y). The repeated multiplication method, although simpler conceptually, can become quite tedious for larger exponents. Choose the method that suits your understanding and comfort level.