(n%2B3)(n-1).jpeg "(n+5)(n+3)(n-1)")
Exploring the Expression (n+5)(n+3)(n-1)
The expression (n+5)(n+3)(n-1) represents the product of three linear factors. Let's explore its properties and how we can work with it.
Expanding the Expression
We can expand the expression by applying the distributive property (often referred to as FOIL for first, outer, inner, last) multiple times. Here's how:
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Expand (n+5)(n+3): (n+5)(n+3) = n(n+3) + 5(n+3) = n² + 3n + 5n + 15 = n² + 8n + 15
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Multiply the result by (n-1): (n² + 8n + 15)(n-1) = n²(n-1) + 8n(n-1) + 15(n-1) = n³ - n² + 8n² - 8n + 15n - 15
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Simplify the expression: n³ - n² + 8n² - 8n + 15n - 15 = n³ + 7n² + 7n - 15
Therefore, the expanded form of (n+5)(n+3)(n-1) is n³ + 7n² + 7n - 15.
Finding the Roots
The roots of the expression are the values of 'n' that make the expression equal to zero. We can find them by setting the expanded form equal to zero and solving the equation:
n³ + 7n² + 7n - 15 = 0
While finding the exact roots of this cubic equation might require numerical methods, we can use the factored form of the expression to easily find one root:
(n+5)(n+3)(n-1) = 0
This equation holds true when any of the factors are equal to zero. Therefore, the roots of the expression are:
- n = -5
- n = -3
- n = 1
Applications
This expression can be used in various applications, including:
- Modeling polynomial functions: It represents a polynomial function of degree 3, which can be used to model real-world phenomena.
- Solving cubic equations: By setting the expression equal to zero, we obtain a cubic equation that can be solved to find its roots.
- Graphing functions: The expression can be used to plot the graph of the corresponding function.
Conclusion
The expression (n+5)(n+3)(n-1) represents a cubic polynomial. Expanding it gives us the form n³ + 7n² + 7n - 15, which can be used to find the roots of the equation and for various other applications. Understanding this expression helps us explore the world of polynomials and their applications in different fields.