Exploring the Expression (x+1)(x+2)(x+3)(x+4)15
This article will explore the fascinating properties of the expression (x+1)(x+2)(x+3)(x+4)15. We will analyze its structure, discover its roots, and examine some of its interesting features.
Understanding the Structure
The expression is a polynomial of degree 4. It can be expanded to get:
(x+1)(x+2)(x+3)(x+4)15 = x⁴ + 10x³ + 35x² + 50x + 21
The expression is also a difference of two terms: the product of four consecutive terms and a constant, 15.
Finding the Roots
The roots of the polynomial are the values of x that make the expression equal to zero. Finding these roots can be a bit challenging directly, but we can use some algebraic manipulations to simplify the problem:

Recognize the pattern: The product of the first four consecutive terms, (x+1)(x+2)(x+3)(x+4), is a special case of the product of consecutive terms, which can be represented by a factorial function.

Rewrite the expression: We can rewrite the expression as: (x+1)(x+2)(x+3)(x+4)  15 = (x+1)(x+2)(x+3)(x+4)  3*5

Factor out common terms: Notice that 3 and 5 can be factored out from the original expression, leaving: (x+1)(x+2)(x+3)(x+4)  35 = (x+1)(x+2)(x+3)(x+4)  (x+1)(x+2)(x+3) + (x+1)(x+2)(x+3)  35

Group the terms: We can group the terms as: [(x+1)(x+2)(x+3)(x+4)  (x+1)(x+2)(x+3)] + [(x+1)(x+2)(x+3)  3*5]

Factor by grouping: Factor out common terms from each group: (x+1)(x+2)(x+3) *(x+4  1) + (x+1)(x+2) *(x+3  5)

Simplify: (x+1)(x+2)(x+3)(x+3) + (x+1)(x+2)(x2)

Factor out (x+1)(x+2): (x+1)(x+2) * [(x+3)(x+3) + (x2)]

Expand and simplify: (x+1)(x+2) * (x² + 6x + 9 + x  2)

Final form: (x+1)(x+2) * (x² + 7x + 7)
This gives us the factored form of the expression. Now to find the roots, we set the expression equal to zero and solve for x:
 (x+1)(x+2) * (x² + 7x + 7) = 0
This equation has solutions when any of the factors equal zero:
 x + 1 = 0 => x = 1
 x + 2 = 0 => x = 2
 x² + 7x + 7 = 0
The last equation can be solved using the quadratic formula, leading to two more solutions for x.
Further Exploration
The expression (x+1)(x+2)(x+3)(x+4)15 has some intriguing properties:
 Symmetry: The expression exhibits symmetry when rearranged as (x+1)(x+4)(x+2)(x+3)15. This is due to the arrangement of consecutive terms.
 Relationship to factorials: The product of four consecutive terms, (x+1)(x+2)(x+3)(x+4), can be represented in terms of factorials. This connection reveals interesting patterns and relationships with other mathematical concepts.
 Applications: Expressions similar to this one find applications in various fields like algebra, calculus, and physics.
This expression presents a great opportunity to explore the fascinating world of algebra and polynomial functions. By understanding its structure, roots, and patterns, we gain deeper insights into the beauty and power of mathematics.