(x-5)(x-7)(x-9)-9.jpeg "(x-3)(x-5)(x-7)(x-9)-9")
Exploring the Expression (x-3)(x-5)(x-7)(x-9) - 9
This article will delve into the intriguing expression (x-3)(x-5)(x-7)(x-9) - 9. We will analyze its structure, explore its properties, and discover some interesting insights.
Understanding the Expression
The expression consists of two main parts:
- Product of linear factors: (x-3)(x-5)(x-7)(x-9) represents the product of four linear expressions. Each factor is in the form of (x-a), where 'a' is a constant.
- Constant term: -9 is a constant term subtracted from the product of linear factors.
Expanding the Expression
To better understand the expression, we can expand it by multiplying out the factors. Here's how:
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First two factors: (x-3)(x-5) = x² - 8x + 15
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Next two factors: (x-7)(x-9) = x² - 16x + 63
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Multiplying the results: (x² - 8x + 15)(x² - 16x + 63) = x⁴ - 24x³ + 203x² - 736x + 945
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Subtracting the constant term: x⁴ - 24x³ + 203x² - 736x + 945 - 9 = x⁴ - 24x³ + 203x² - 736x + 936
Therefore, the expanded form of the expression is x⁴ - 24x³ + 203x² - 736x + 936.
Properties of the Expression
- Degree: The expression is a fourth-degree polynomial, as the highest power of x is 4.
- Roots: The expression has four roots, which are the values of x that make the expression equal to zero. These roots are 3, 5, 7, and 9. This is directly evident from the factored form of the expression, where each factor becomes zero when x equals its corresponding root.
- Symmetry: The expression is not symmetrical.
- Behavior at Infinity: As x approaches positive or negative infinity, the expression tends towards positive infinity due to the dominance of the x⁴ term.
Conclusion
The expression (x-3)(x-5)(x-7)(x-9) - 9 is a fourth-degree polynomial with four roots, representing a specific curve on a graph. Analyzing its structure and expanding it provides valuable insights into its behavior and properties. Further exploration might involve examining its graph, finding its critical points, or exploring its relationships to other mathematical concepts.