(x%2B1)(x%2B3).jpeg "(x-8)(x+1)(x+3)")
Exploring the Polynomial (x-8)(x+1)(x+3)
This article will delve into the polynomial expression (x-8)(x+1)(x+3). We will explore its factored form, expand it to its standard form, identify its key features, and discuss its applications.
Factored Form:
The expression is already in its factored form, which is a product of three linear factors:
- (x-8): This factor indicates a root of the polynomial at x=8.
- (x+1): This factor indicates a root of the polynomial at x=-1.
- (x+3): This factor indicates a root of the polynomial at x=-3.
Expanded Form:
To obtain the standard form of the polynomial, we need to expand the expression. This can be done by applying the distributive property multiple times.
Step 1: Expand (x-8)(x+1) (x-8)(x+1) = x(x+1) - 8(x+1) = x² + x - 8x - 8 = x² - 7x - 8
Step 2: Multiply the result from step 1 by (x+3) (x² - 7x - 8)(x+3) = x²(x+3) - 7x(x+3) - 8(x+3) = x³ + 3x² - 7x² - 21x - 8x - 24
Step 3: Simplify the expression x³ + 3x² - 7x² - 21x - 8x - 24 = x³ - 4x² - 29x - 24
Therefore, the standard form of the polynomial is x³ - 4x² - 29x - 24.
Key Features:
- Degree: The highest power of x in the polynomial is 3, indicating it is a cubic polynomial.
- Roots: As we identified in the factored form, the polynomial has three roots: x = 8, x = -1, and x = -3.
- Y-intercept: The y-intercept is found by setting x = 0 in the standard form of the polynomial. In this case, the y-intercept is (0, -24).
Applications:
Polynomials like this are used in various fields, including:
- Engineering: Modeling physical phenomena like the motion of objects, fluid flow, and electrical circuits.
- Economics: Analyzing economic data, predicting market trends, and developing economic models.
- Computer Science: Representing curves and surfaces in computer graphics, and designing algorithms for numerical analysis.
Conclusion:
The polynomial (x-8)(x+1)(x+3) is a cubic polynomial with three distinct roots. We explored its factored and expanded forms, identified its key features, and highlighted some of its applications. Understanding these concepts can be helpful in various academic and professional fields.