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(x+1)(x+2)(x+3)(x+4)-15

6 min read Jun 16, 2024
(x+1)(x+2)(x+3)(x+4)-15

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:

  1. 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.

  2. 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

  3. 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

  4. 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]

  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)

  6. Simplify: (x+1)(x+2)(x+3)(x+3) + (x+1)(x+2)(x-2)

  7. Factor out (x+1)(x+2): (x+1)(x+2) * [(x+3)(x+3) + (x-2)]

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

  9. 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.

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