(x%2B4)(x%2B5)(x%2B6)%2B1.jpeg "(x+3)(x+4)(x+5)(x+6)+1")
Factoring and Exploring (x+3)(x+4)(x+5)(x+6)+1
The expression (x+3)(x+4)(x+5)(x+6)+1 might seem complex at first glance, but with some clever manipulation, we can uncover its hidden beauty and explore its fascinating properties.
Factoring by Grouping
The key to factoring this expression lies in grouping the terms strategically. Let's rearrange the terms and observe:
(x+3)(x+6) * (x+4)(x+5) + 1
Notice that the first two factors and the last two factors share a common pattern. This allows us to apply the difference of squares factorization:
[(x+3)(x+6) + 1][(x+4)(x+5) + 1] - 1
Now, we expand the products inside the brackets:
[(x² + 9x + 18) + 1][(x² + 9x + 20) + 1] - 1
This simplifies to:
(x² + 9x + 19)(x² + 9x + 21) - 1
Final Factored Form
Finally, we can apply the difference of squares factorization one more time to obtain the fully factored form:
[(x² + 9x + 20) - 1][(x² + 9x + 20) + 1]
This simplifies to:
(x² + 9x + 19)(x² + 9x + 21)
Exploring the Properties
We have successfully factored the expression. This factored form reveals some interesting properties:
- Symmetry: Notice the symmetry in the factors. Both factors are quadratic expressions with the same linear term (9x). This symmetry suggests potential connections to other mathematical concepts.
- Roots: Finding the roots of the expression involves setting each factor equal to zero and solving the resulting quadratic equations. This will give us four solutions for the original expression.
- Graphing: The factored form can be used to graph the function represented by the expression. The roots will correspond to the x-intercepts of the graph, and the symmetry will be evident in the shape of the curve.
Conclusion
By strategically factoring and manipulating the expression (x+3)(x+4)(x+5)(x+6)+1, we have uncovered its fascinating structure and gained insights into its properties. This journey showcases the power of algebraic manipulation and the beauty of mathematical patterns.