(x-2)(x-7).jpeg "(x+1)(x-2)(x-7)")
Exploring the Polynomial (x+1)(x-2)(x-7)
This article explores the polynomial (x+1)(x-2)(x-7), focusing on its properties and how to work with it.
Understanding the Factorization
The polynomial (x+1)(x-2)(x-7) is presented in factored form, which makes it easy to identify its roots (also known as zeros). These are the values of x that make the polynomial equal to zero.
From the factored form, we can immediately see the roots are:
- x = -1
- x = 2
- x = 7
This is because if any of these values are substituted for x, one of the factors will become zero, making the entire product equal to zero.
Expanding the Polynomial
To understand the polynomial's standard form, we can expand the product:
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Start with two factors: (x+1)(x-2) = x² - x - 2
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Multiply the result by the third factor: (x² - x - 2)(x-7) = x³ - 8x² + 9x + 14
Therefore, the standard form of the polynomial is x³ - 8x² + 9x + 14.
Analyzing the Polynomial
Here are some key properties of this polynomial:
- Degree: The highest power of x is 3, making it a cubic polynomial.
- Leading Coefficient: The coefficient of the highest power term (x³) is 1.
- Constant Term: The constant term is 14.
Using the Polynomial
The factored form is particularly useful for:
- Finding the x-intercepts: The x-intercepts are the points where the graph of the polynomial crosses the x-axis. These correspond to the roots of the polynomial, which we already determined as (-1, 0), (2, 0), and (7, 0).
- Solving equations: If we set the polynomial equal to a specific value, we can solve for the corresponding values of x. For example, solving the equation (x+1)(x-2)(x-7) = 0 would yield the same roots as before.
Conclusion
The polynomial (x+1)(x-2)(x-7) is a simple cubic polynomial that can be analyzed and manipulated using its factored form and expanded form. Understanding its properties and how to work with it is essential for solving equations, finding intercepts, and gaining insights into its behavior.