(x-6)(x-7).jpeg "(x-5)(x-6)(x-7)")
Exploring the Cubic Equation (x-5)(x-6)(x-7)
This article delves into the cubic equation (x-5)(x-6)(x-7), exploring its key features, solving methods, and graphical representation.
Understanding the Equation
The equation (x-5)(x-6)(x-7) represents a cubic function, meaning it has a highest power of x as 3. This form is particularly useful as it directly reveals the roots of the equation, which are the values of x that make the equation equal to zero.
Roots:
- x = 5
- x = 6
- x = 7
These roots are easily identifiable since they are the values that make each factor in the equation equal to zero.
Expanding the Equation
To obtain the standard form of the cubic equation, we can expand the product:
(x-5)(x-6)(x-7) = (x² - 11x + 30)(x - 7) = x³ - 18x² + 107x - 210
This expanded form reveals the coefficients of the cubic, quadratic, linear, and constant terms.
Graphical Representation
The graph of this cubic function will intersect the x-axis at the three roots: x = 5, x = 6, and x = 7. The graph will have a general shape similar to a "S" curve, with one turning point between x = 5 and x = 6 and another between x = 6 and x = 7.
Solving the Equation
Since we already identified the roots by factoring the equation, there is no need for additional solving methods. However, if the equation were presented in the standard form (x³ - 18x² + 107x - 210 = 0), we could employ various techniques for solving cubic equations, such as:
- Rational Root Theorem: This theorem helps find potential rational roots.
- Synthetic Division: This method allows us to divide the polynomial by a potential root to check if it is a solution.
- Cubic Formula: This complex formula can be used to find all three roots, though it is often computationally intensive.
Applications
Cubic equations like this one have applications in various fields, including:
- Physics: Modelling projectile motion or oscillations.
- Engineering: Designing structures or analyzing fluid flow.
- Economics: Predicting market trends or analyzing investment strategies.
In Conclusion:
The equation (x-5)(x-6)(x-7) provides a simple example of a cubic function with easily identifiable roots. By understanding its factors, expansion, and graphical representation, we gain insights into the behavior of cubic equations and their potential applications in diverse fields.