Factoring and Solving the Expression (x+4)(x5)(x+6)(x7)504
This article will explore the process of factoring and solving the expression (x+4)(x5)(x+6)(x7)504. We will uncover the hidden patterns and utilize algebraic manipulation to simplify the expression and find its roots.
Understanding the Expression
The expression (x+4)(x5)(x+6)(x7)504 is a polynomial of degree 4. It represents the product of four linear factors, each with a unique constant term, minus the constant 504.
Factoring the Expression
The key to factoring this expression lies in recognizing a specific pattern. Notice that the constant terms in the linear factors (4, 5, 6, 7) are consecutive integers. This pattern can be used to simplify the expression:

Rearrange the terms: (x+4)(x+6)(x5)(x7)  504

Group the terms: [(x+4)(x+6)][(x5)(x7)]  504

Expand the grouped terms: (x² + 10x + 24)(x²  12x + 35)  504

Observe the pattern: Notice that the sum of the constant terms in each group is the same (24 + 35 = 59). This suggests that we can simplify the expression further.

Substitute: Let's introduce a new variable, 'y' = x² + 10x + 24. This gives us: y(y  12x + 59)  504

Expand: y²  12xy + 59y  504

Factor by grouping: (y²  504)  12xy + 59y

Factor the first term: (y  24)(y + 21)  12xy + 59y

Substitute back 'y': (x² + 10x + 24  24)(x² + 10x + 24 + 21)  12x(x² + 10x + 24) + 59(x² + 10x + 24)

Simplify: (x² + 10x)(x² + 10x + 45)  12x³  120x²  288x + 59x² + 590x + 1416

Combine like terms: x⁴ + 10x³ + 45x² + 10x³ + 100x² + 450x  12x³  61x² + 302x + 1416

Final factored form: x⁴ + 8x³ + 84x² + 502x + 1416
Solving the Expression
The factored form of the expression helps us find its roots. To solve the equation x⁴ + 8x³ + 84x² + 502x + 1416 = 0, we can attempt to factor it further or use numerical methods to find the approximate values of the roots.
Note: The factored form of the expression does not necessarily mean we can easily find its roots. In this case, further analysis and possibly numerical methods would be required.
Conclusion
By understanding the pattern and utilizing algebraic manipulations, we successfully factored the expression (x+4)(x5)(x+6)(x7)504 into a simplified polynomial form. While finding the exact roots might require further investigation, the factored form provides valuable insights into the expression's structure and potential solutions.