(x-3)(x-5)(x-7).jpeg "(x-1)(x-3)(x-5)(x-7)")
Exploring the Polynomial (x-1)(x-3)(x-5)(x-7)
This article will delve into the polynomial (x-1)(x-3)(x-5)(x-7), examining its properties, how to expand it, and its significance in mathematics.
Understanding the Structure
The polynomial (x-1)(x-3)(x-5)(x-7) is presented in its factored form. This form reveals crucial information about the polynomial:
- Roots: The factors (x-1), (x-3), (x-5), and (x-7) directly indicate that the polynomial has roots at x = 1, x = 3, x = 5, and x = 7. These are the values of x for which the polynomial evaluates to zero.
- Degree: The polynomial has a degree of 4, as it is the product of four linear factors. This means it is a quartic polynomial.
Expanding the Polynomial
To understand the polynomial's behavior more fully, we can expand it:
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First, multiply the first two factors: (x-1)(x-3) = x² - 4x + 3
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Then, multiply the last two factors: (x-5)(x-7) = x² - 12x + 35
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Finally, multiply the results from steps 1 and 2: (x² - 4x + 3)(x² - 12x + 35) = x⁴ - 16x³ + 83x² - 172x + 105
Therefore, the expanded form of the polynomial is:
x⁴ - 16x³ + 83x² - 172x + 105
Significance and Applications
This particular polynomial, while seemingly simple, has significance in various mathematical contexts:
- Function Analysis: Understanding its roots and degree allows us to analyze the function's behavior: its intercepts, turning points, and long-term trends.
- Polynomial Equation Solving: The factored form helps solve the equation (x-1)(x-3)(x-5)(x-7) = 0, directly providing the solutions x = 1, x = 3, x = 5, and x = 7.
- Curve Fitting: This polynomial can be used to model data points in certain scenarios, providing a smooth curve that passes through the points.
Conclusion
The polynomial (x-1)(x-3)(x-5)(x-7) provides a rich example for exploring the concepts of roots, degree, expansion, and the relationship between a polynomial's factored and expanded forms. Its applications extend beyond basic algebra, making it a valuable tool in various mathematical fields.