(x-2)(x-8).jpeg "(x+3)(x-2)(x-8)")
Factoring and Expanding the Expression (x+3)(x-2)(x-8)
This expression represents a polynomial that is already factored. Let's explore how to expand it and understand its properties.
Expanding the Expression
To expand the expression, we need to multiply the factors together:
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Start with the first two factors: (x+3)(x-2) = x² + x - 6
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Now multiply the result by the third factor: (x² + x - 6)(x-8) = x³ - 7x² - 50x + 48
Therefore, the expanded form of the expression is x³ - 7x² - 50x + 48.
Understanding the Expression
- Degree: The highest power of x in the expression is 3, making it a third-degree polynomial or a cubic polynomial.
- Roots: The factors of the expression represent the roots (or zeros) of the polynomial. When you set the polynomial equal to zero and solve for x, you'll find that the roots are:
- x = -3
- x = 2
- x = 8
Graphing the Polynomial
The graph of this polynomial will intersect the x-axis at the points (-3, 0), (2, 0), and (8, 0). The shape of the graph will be a curve with a general "S" shape, characteristic of a cubic polynomial.
Applications
Understanding how to expand and factor polynomials is essential in various fields, including:
- Algebra: Solving equations, finding roots, and understanding the behavior of functions.
- Calculus: Finding derivatives and integrals, and analyzing the behavior of curves.
- Physics: Modeling physical phenomena, such as projectile motion and wave patterns.
- Engineering: Designing structures, analyzing systems, and solving optimization problems.
By mastering the techniques of factoring and expanding polynomials, you gain a powerful tool for understanding and manipulating mathematical expressions.