(x%2B3)(x%2B5)(x%2B7)(x%2B9)(x%2B11)(x%2B13)(x%2B15).jpeg "(x+1)(x+3)(x+5)(x+7)(x+9)(x+11)(x+13)(x+15)")
Exploring the Expansion of (x+1)(x+3)(x+5)(x+7)(x+9)(x+11)(x+13)(x+15)
This expression represents the product of eight consecutive odd numbers, each increased by 1. While the full expansion of this expression is quite lengthy and complex, we can explore some interesting aspects and patterns related to it.
Understanding the Structure
The given expression is a product of linear factors. Each factor (x + 1), (x + 3), etc., represents a line on a graph. Multiplying these factors together creates a polynomial function. The degree of this polynomial is 8, meaning it will have a maximum of 8 roots (values of x where the function equals zero).
Key Properties
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Roots: The expression equals zero when any of the factors equals zero. This means the roots of the polynomial are -1, -3, -5, -7, -9, -11, -13, and -15.
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Symmetry: Due to the pattern of consecutive odd numbers, the roots are symmetrically placed around the y-axis.
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Leading Coefficient: The coefficient of the highest power of x (x⁸) will be 1, as the leading coefficient of each factor is 1.
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Constant Term: The constant term of the expanded polynomial will be the product of all the constant terms in the factors, which is 1 * 3 * 5 * 7 * 9 * 11 * 13 * 15.
Expansion and Patterns
While expanding the entire expression manually would be tedious, we can observe some patterns:
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Coefficients: The coefficients of the expanded polynomial will follow a specific pattern related to the combinations of the constant terms in the factors. For instance, the coefficient of x⁷ will be the sum of all possible products of seven of the constant terms (1, 3, 5, etc.).
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Symmetry: The coefficients of the polynomial will be symmetric around the middle term (x⁴). This is due to the symmetry of the roots.
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Factoring by Grouping: The expression could be factored by grouping to simplify the expansion, but this would still involve a considerable number of steps.
Conclusion
The expression (x+1)(x+3)(x+5)(x+7)(x+9)(x+11)(x+13)(x+15) represents a polynomial with specific properties related to its roots, symmetry, and coefficients. While a full expansion would be intricate, understanding the structure and patterns allows for insightful analysis of this intriguing mathematical construct.