%5E4-expand.jpeg "(x+5)^4 Expand")
Expanding (x+5)^4: A Step-by-Step Guide
Expanding expressions like (x+5)^4 can seem daunting, but it's actually a straightforward process using the binomial theorem. Here's a breakdown of how to do it:
Understanding the Binomial Theorem
The binomial theorem provides a formula for expanding any expression of the form (a+b)^n:
(a+b)^n = ∑(n choose k) * a^(n-k) * b^k
Where:
- n is the power of the binomial
- k is the index of summation, ranging from 0 to n
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!)
Expanding (x+5)^4
Let's apply the binomial theorem to our example, (x+5)^4:
- Identify 'a' and 'b': In this case, a = x and b = 5.
- Determine 'n': n = 4.
Now, let's expand the expression using the formula:
(x + 5)^4 = ∑(4 choose k) * x^(4-k) * 5^k
We need to calculate the terms for k = 0, 1, 2, 3, and 4:
- k = 0: (4 choose 0) * x^4 * 5^0 = 1 * x^4 * 1 = x^4
- k = 1: (4 choose 1) * x^3 * 5^1 = 4 * x^3 * 5 = 20x^3
- k = 2: (4 choose 2) * x^2 * 5^2 = 6 * x^2 * 25 = 150x^2
- k = 3: (4 choose 3) * x^1 * 5^3 = 4 * x * 125 = 500x
- k = 4: (4 choose 4) * x^0 * 5^4 = 1 * 1 * 625 = 625
Finally, add all the terms together:
(x + 5)^4 = x^4 + 20x^3 + 150x^2 + 500x + 625
Conclusion
By understanding the binomial theorem and following the steps outlined above, you can expand any binomial expression, no matter how complex it may seem. Remember to use the formula and calculate the individual terms carefully.