(x%2B7)(x-4).jpeg "(x-3)(x+7)(x-4)")
Factoring and Solving the Polynomial (x-3)(x+7)(x-4)
This expression represents a polynomial in factored form. Let's explore its properties and how to solve it.
Understanding the Factored Form
The expression (x-3)(x+7)(x-4) is a product of three binomials. Each binomial represents a linear factor:
- (x-3): This factor indicates that the polynomial has a root (or zero) at x = 3.
- (x+7): This factor indicates a root at x = -7.
- (x-4): This factor indicates a root at x = 4.
Expanding the Expression
To obtain the polynomial in its standard form, we need to expand the factored expression:
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Multiply the first two binomials: (x-3)(x+7) = x² + 4x - 21
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Multiply the result by the third binomial: (x² + 4x - 21)(x-4) = x³ - 4x² + 4x² - 16x - 21x + 84
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Simplify: x³ - 37x + 84
Therefore, the expanded form of the polynomial is x³ - 37x + 84.
Finding the Roots
As we saw earlier, the factored form readily provides the roots of the polynomial: x = 3, x = -7, and x = 4.
These roots are also the x-intercepts of the polynomial's graph.
Applications
Understanding factored forms and roots of polynomials is essential in various mathematical and scientific applications, such as:
- Solving equations: Finding the values of x that make the polynomial equal to zero.
- Graphing functions: Identifying the x-intercepts and behavior of the graph.
- Optimization problems: Determining maximum and minimum values of functions.
- Modeling real-world phenomena: Representing relationships between variables in physics, engineering, and economics.
Conclusion
The expression (x-3)(x+7)(x-4) represents a polynomial in factored form, providing a clear understanding of its roots and behavior. Expanding the expression leads to its standard form, which is useful for further analysis and solving equations. Understanding these concepts is crucial in various mathematical and scientific applications.