Cubic Polynomial Function Equation With Given Zeroes And Y-Intercept
Suppose a cubic polynomial function has the same zeroes and passes through the coordinate (0,-5). Write the equation of this cubic polynomial function, given that the zeroes are (2,0), (3,0), and (5,0). What is the y-intercept?
In this exploration, we delve into the fascinating world of polynomial functions, specifically focusing on cubic polynomial functions. Our primary objective is to construct the equation of a cubic polynomial function given its zeroes and a point it passes through. This exercise showcases the fundamental relationship between the roots of a polynomial and its algebraic representation. We will use the zeroes (2,0), (3,0), and (5,0), and the point (0,-5) to derive the unique cubic polynomial function that satisfies these conditions. Understanding how to build such equations is crucial in various fields, including engineering, physics, and computer science, where polynomial models are frequently used to represent real-world phenomena. This exploration will not only reinforce your understanding of polynomial functions but also demonstrate the practical application of algebraic concepts in problem-solving.
The zeroes of a polynomial function are the x-values where the function equals zero, and these values correspond to the x-intercepts of the graph. Knowing the zeroes of a polynomial is a powerful tool, as it allows us to express the polynomial in factored form. This form is particularly useful for identifying the polynomial's behavior and for constructing its equation. The y-intercept, on the other hand, is the point where the graph intersects the y-axis, providing another crucial piece of information about the polynomial's vertical position. By combining the information about the zeroes and the y-intercept, we can uniquely determine the cubic polynomial function.
Before diving into the problem, let's clarify some core concepts about cubic polynomial functions. A cubic polynomial function is a polynomial function of degree three, which means the highest power of the variable x is 3. The general form of a cubic polynomial is given by:
f(x) = ax^3 + bx^2 + cx + d
where a, b, c, and d are constants, and a ≠ 0. The zeroes of a polynomial function are the values of x for which f(x) = 0. These zeroes are also known as the roots of the polynomial equation. For a cubic polynomial, there can be up to three real roots. In our case, we are given three distinct real roots: 2, 3, and 5. This simplifies the process of constructing the polynomial, as we can directly use the factored form.
The factored form of a polynomial is an expression of the polynomial as a product of linear factors, each corresponding to a root. If r is a root of the polynomial, then (x - r) is a factor. Given the roots 2, 3, and 5, the factored form of our cubic polynomial will look like:
f(x) = k(x - 2)(x - 3)(x - 5)
where k is a constant. This constant is crucial because it scales the polynomial vertically, affecting its y-intercept. To find the specific value of k, we need additional information, which is provided in the problem as the point (0, -5). This point represents the y-intercept of the graph, and it will allow us to determine the exact cubic polynomial function. Understanding the factored form and the role of the leading coefficient k is essential for constructing polynomial equations from their roots and other given conditions.
To construct the cubic polynomial function, we start with the factored form, which we've already established as:
f(x) = k(x - 2)(x - 3)(x - 5)
Here, k is the constant that we need to determine. To find k, we use the given point (0, -5), which lies on the graph of the function. This means that when x = 0, f(x) = -5. Substituting these values into the equation, we get:
-5 = k(0 - 2)(0 - 3)(0 - 5)
Simplifying the expression:
-5 = k(-2)(-3)(-5) -5 = k(-30)
Now, we solve for k:
k = -5 / -30 k = 1/6
Now that we have the value of k, we can write the complete factored form of the cubic polynomial function:
f(x) = (1/6)(x - 2)(x - 3)(x - 5)
To express the polynomial in the standard form (ax^3 + bx^2 + cx + d), we need to expand the factored form. This involves multiplying the factors together and then distributing the constant (1/6). This process will give us the coefficients of the polynomial in its standard form. Expanding the factored form is a crucial step in fully understanding the polynomial's behavior and its algebraic representation.
Expanding the factored form of the cubic polynomial involves a systematic multiplication of the factors. We start with:
f(x) = (1/6)(x - 2)(x - 3)(x - 5)
First, let's multiply the first two factors:
(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6
Now, we multiply the result by the third factor (x - 5):
(x^2 - 5x + 6)(x - 5) = x^3 - 5x^2 - 5x^2 + 25x + 6x - 30 = x^3 - 10x^2 + 31x - 30
Finally, we distribute the constant (1/6) across the resulting polynomial:
f(x) = (1/6)(x^3 - 10x^2 + 31x - 30) f(x) = (1/6)x^3 - (10/6)x^2 + (31/6)x - (30/6) f(x) = (1/6)x^3 - (5/3)x^2 + (31/6)x - 5
This is the cubic polynomial function in standard form. Each term's coefficient provides valuable information about the polynomial's shape and behavior. The leading coefficient (1/6) indicates the end behavior of the graph, and the other coefficients influence the curve's shape and position. This expanded form allows for easier analysis of the polynomial's properties and behavior. It's a critical step in fully characterizing the cubic polynomial function.
The y-intercept of a function is the point where the graph intersects the y-axis. This occurs when x = 0. We have already been given that the polynomial passes through the point (0, -5), which directly tells us that the y-intercept is -5. However, let's confirm this using the expanded form of the polynomial:
f(x) = (1/6)x^3 - (5/3)x^2 + (31/6)x - 5
To find the y-intercept, we substitute x = 0 into the function:
f(0) = (1/6)(0)^3 - (5/3)(0)^2 + (31/6)(0) - 5 f(0) = 0 - 0 + 0 - 5 f(0) = -5
This confirms that the y-intercept is indeed -5, which corresponds to the point (0, -5). The y-intercept is a crucial feature of the polynomial as it represents the value of the function when x is zero. It's the constant term in the polynomial's standard form, which in this case is -5. Understanding the y-intercept is essential for sketching the graph of the polynomial and for analyzing its behavior. In real-world applications, the y-intercept can represent an initial condition or a starting value in a model represented by the polynomial function. Verifying the y-intercept using both the given point and the expanded form reinforces our understanding of the polynomial's properties and behavior.
In this detailed exploration, we have successfully constructed the equation of a cubic polynomial function given its zeroes and a point it passes through. We began by understanding the general and factored forms of cubic polynomials. Using the zeroes (2, 0), (3, 0), and (5, 0), we established the factored form of the polynomial as f(x) = k(x - 2)(x - 3)(x - 5). The critical step was determining the constant k, which we found to be 1/6 by using the given point (0, -5). This point represents the y-intercept and allowed us to solve for k.
Next, we expanded the factored form to obtain the polynomial in its standard form: f(x) = (1/6)x^3 - (5/3)x^2 + (31/6)x - 5. This standard form provides a clear representation of the polynomial's coefficients, which influence its shape and behavior. We verified that the y-intercept is indeed -5 by substituting x = 0 into the expanded form, confirming our calculations and understanding of the polynomial's properties.
This exercise demonstrates the powerful connection between the roots of a polynomial and its equation. By understanding the factored and standard forms, we can construct polynomials that meet specific criteria. The ability to determine polynomial equations from given conditions is a fundamental skill in mathematics and has broad applications in various fields, including engineering, physics, and computer science. The process of constructing and analyzing polynomials reinforces our understanding of algebraic concepts and their practical implications in modeling real-world phenomena.