(x%2B2)(x%2B3)(x%2B4)(x%2B5)-simplify.jpeg "(x+1)(x+2)(x+3)(x+4)(x+5) Simplify")
Simplifying the Expression (x+1)(x+2)(x+3)(x+4)(x+5)
This expression represents the product of five consecutive factors. While expanding it directly would be quite tedious, we can use a clever trick to simplify it.
The Trick
The key is to recognize that this expression resembles a pattern found in factorials. Factorials are defined as the product of consecutive integers starting from 1 down to a given number. For example, 5! = 5 * 4 * 3 * 2 * 1.
We can rewrite our expression as:
(x+5)! / x!
This is because the product (x+1)(x+2)(x+3)(x+4)(x+5) represents the consecutive integers from (x+1) to (x+5), and dividing by x! eliminates the terms from 1 to x.
Further Simplification
While this is a simplified form, it can be further expanded for specific values of x. For example, if x = 2, the expression becomes:
7! / 2! = 5040 / 2 = 2520
Conclusion
Simplifying the expression (x+1)(x+2)(x+3)(x+4)(x+5) can be achieved by recognizing the pattern of factorials and rewriting it as (x+5)! / x!. This form is more concise and easier to work with, especially when dealing with specific values of x.