Exploring the Expression (x+1)(x7)(x+3)
This article will delve into the expression (x+1)(x7)(x+3), analyzing its properties and exploring ways to work with it.
Understanding the Expression
The expression represents a product of three linear factors:
 (x+1): This factor will equal zero when x = 1.
 (x7): This factor will equal zero when x = 7.
 (x+3): This factor will equal zero when x = 3.
Expanding the Expression
We can expand the expression to obtain a polynomial:

First, expand (x+1)(x7): (x+1)(x7) = x²  7x + x  7 = x²  6x  7

Then, multiply the result by (x+3): (x²  6x  7)(x+3) = x³ + 3x²  6x²  18x  7x  21

Simplify the expression: (x+1)(x7)(x+3) = x³  3x²  25x  21
Finding the Roots
The roots of the expression are the values of x that make the expression equal to zero. Since the expression is a product of three factors, we know that it will be zero if any of the factors are zero.
Therefore, the roots are:
 x = 1
 x = 7
 x = 3
Applications
This expression, in its factored or expanded form, can be used in various applications, including:
 Solving equations: If we set the expression equal to a constant, we can solve for the values of x that satisfy the equation.
 Graphing functions: The expression represents a cubic function. Understanding its factored form helps visualize the graph, including its intercepts and turning points.
 Calculus: The expression can be used to find the derivative and integral of the corresponding function.
Conclusion
The expression (x+1)(x7)(x+3) represents a cubic function with roots at 1, 7, and 3. It can be expanded into a polynomial form, providing insights into its properties and applications in various mathematical contexts.