(x-7)(x%2B3).jpeg "(x+1)(x-7)(x+3)")
Exploring the Expression (x+1)(x-7)(x+3)
This article will delve into the expression (x+1)(x-7)(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.
- (x-7): 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:
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First, expand (x+1)(x-7): (x+1)(x-7) = x² - 7x + x - 7 = x² - 6x - 7
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Then, multiply the result by (x+3): (x² - 6x - 7)(x+3) = x³ + 3x² - 6x² - 18x - 7x - 21
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Simplify the expression: (x+1)(x-7)(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)(x-7)(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.