In the realm of mathematics, the concept of function zeros—also known as roots—holds significant importance. These points where a function intersects the x-axis reveal valuable information about the behavior and characteristics of that function. To address the question “Which function has zeros at x = 10 and x = 2?”, we will delve into the construction and properties of polynomials, specifically focusing on how to derive a function that meets these criteria.
Initially, it is imperative to comprehend what it means for a function to have zeros. A function ( f(x) ) possesses a zero at a specific point ( x = a ) if ( f(a) = 0 ). In our case, the function must yield zero when ( x ) equals both 10 and 2. This condition underscores a wider mathematical phenomenon, where solutions to equations often yield insights into the underlying structure of the function itself.
To construct a polynomial function with designated zeros, we typically employ these zeros in a linear factored form. For our specific zeros, x = 10 and x = 2, we can express the function as:
( f(x) = (x – 10)(x – 2) )
This expression signifies that our polynomial will have two fundamental linear factors, each corresponding to one of the zeros. A polynomial’s degree is fundamentally linked to the number of its roots; thus, the resulting function is a quadratic polynomial, given that it is formed by multiplying two linear factors.
Upon expanding this factored form, we unveil a clearer representation of the function:
( f(x) = x^2 – 12x + 20 )
Here, we have seamlessly transitioned from the factored version to the standard form of a quadratic equation. The terms produced give additional insights into the nature of the polynomial. The leading coefficient, which in this case is 1, indicates the function opens upwards, revealing that it reaches a minimum value rather than a maximum.
The intricate binding of zeros to their polynomial habitat leads to several intriguing observations about the graph of the function. As we plot ( f(x) ), we will notice two distinct points on the x-axis where the curve intersects—precisely at ( x = 10 ) and ( x = 2 ). More so, the vertex of this quadratic function can be discerned utilizing the vertex formula ( x = -frac{b}{2a} ), where ( b ) is the coefficient of ( x ) and ( a ) is the leading coefficient. In our case:
( x = -frac{-12}{2 times 1} = 6 )
Thus, the x-coordinate of the vertex, x = 6, lies equidistant from our two zeros, showcasing the symmetric nature intrinsic to parabolic functions. The implications of this symmetric property provide a rich tapestry for interpretation, facilitating the study and application of quadratics in diverse fields, from physics to economics.
Furthermore, we can derive some interesting characteristics of our function through the analysis of its coefficients, resulting in the identification of additional features. The sum of the roots can be calculated as:
( r_1 + r_2 = 10 + 2 = 12 )
And the product of the roots manifests as:
( r_1 cdot r_2 = 10 cdot 2 = 20 )
These relationships, articulated in Vieta’s formulas, reinforce the interconnected nature of polynomial equations, linking the roots directly to the coefficients of the polynomial itself. Such equations are not merely numerical constructs but rather gateways to understanding deeper mathematical relationships.
As we transition from the mechanics of deriving and analyzing our polynomial, it becomes pertinent to explore the broader implications of its zeros. Zero points, in various contexts, signal equilibrium or balance. For instance, in physics, they might correspond to points of equilibrium in forces, while in economics, they can indicate break-even points in profit and loss analysis. Thus, zeros of functions serve not only as mathematical curiosities but rather as essential tools for problem-solving across disciplines.
In conclusion, the function that has zeros at ( x = 10 ) and ( x = 2 ) is characterized by the polynomial:
( f(x) = x^2 – 12x + 20 )
The exploration of this function transcends the mere identification of zeros; it encapsulates a journey through the interconnections of algebraic principles, graphical interpretations, and real-world applications. The alluring nature of these mathematical constructs is a testament to their indispensability in fostering a more profound understanding of various phenomena, reinforcing the notion that even in their abstract form, they serve as fundamental tools for deciphering the complexities of the universe.
