Determining A Polynomial Function From Intercepts And A Point
In the realm of polynomial functions, a fascinating problem arises when we are tasked with identifying the specific function that satisfies given conditions. These conditions often include the x-intercepts, which are the points where the graph of the function crosses the x-axis, and a particular point that the graph passes through. This article delves into the process of determining such a polynomial function. We will walk through a step-by-step approach to construct the polynomial, incorporating the provided x-intercepts and using the given point to solve for any necessary coefficients. This exploration is not just a mathematical exercise; it's a critical skill in various fields, including engineering, physics, and computer graphics, where polynomial functions are used to model curves and trajectories. By understanding how to build these functions from their roots and a point, we gain a powerful tool for describing and predicting real-world phenomena.
Before diving into the solution, let's establish a firm grasp on polynomial functions and their relationship with x-intercepts. A polynomial function is a function that can be expressed in the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₁, a₀ are constants (coefficients) and n is a non-negative integer (the degree of the polynomial). The x-intercepts, also known as roots or zeros, are the values of x for which the function f(x) equals zero. Graphically, these are the points where the polynomial's graph intersects the x-axis.
The fundamental theorem of algebra tells us that a polynomial of degree n has n complex roots (counting multiplicities). This means that if we know the x-intercepts of a polynomial, we can begin to construct its factored form. For example, if a polynomial has x-intercepts at x = a, x = b, and x = c, then the polynomial can be written in the form f(x) = k(x - a)(x - b)(x - c), where k is a constant coefficient. This constant, k, stretches or compresses the graph vertically and can flip it over the x-axis if it's negative. To uniquely determine the polynomial, we need additional information, which is often provided in the form of a point that the graph passes through.
The problem at hand presents us with a polynomial function that has specific x-intercepts and passes through a given point. Our goal is to identify which of the provided functions could represent this graph. The given x-intercepts are 3, 0, and -1, and the point the polynomial passes through is (1, -8). This means that when x = 3, x = 0, or x = -1, the function's value is 0, and when x = 1, the function's value is -8. This information allows us to construct a general form of the polynomial and then use the point (1, -8) to solve for any unknown coefficients. By comparing our derived polynomial with the given options, we can pinpoint the function that matches the specified criteria.
Given the x-intercepts 3, 0, and -1, we can express the polynomial function in its factored form. Since the intercepts are the roots of the polynomial, we know that (x - 3), (x - 0), and (x - (-1)) are factors of the polynomial. Thus, the polynomial can be written in the form:
f(x) = k(x - 3)(x)(x + 1)
where k is a constant coefficient that we need to determine. This coefficient accounts for any vertical stretching, compression, or reflection of the graph. To find the value of k, we use the additional information that the polynomial passes through the point (1, -8). This means that when x = 1, f(x) = -8. Substituting these values into our equation, we get:
-8 = k(1 - 3)(1)(1 + 1)
Simplifying this equation, we have:
-8 = k(-2)(1)(2) -8 = -4k
Dividing both sides by -4, we find:
k = 2
Now that we have found the value of k, we can write the complete polynomial function:
f(x) = 2(x - 3)(x)(x + 1)
To compare our constructed polynomial with the given options, we need to expand the factored form into its standard form. Starting with the expression:
f(x) = 2(x - 3)(x)(x + 1)
First, let's multiply the factors (x) and (x + 1):
x(x + 1) = x² + x
Now, we substitute this back into our equation:
f(x) = 2(x - 3)(x² + x)
Next, we multiply (x - 3) by (x² + x):
(x - 3)(x² + x) = x(x² + x) - 3(x² + x) = x³ + x² - 3x² - 3x = x³ - 2x² - 3x
Finally, we multiply the entire expression by 2:
f(x) = 2(x³ - 2x² - 3x) = 2x³ - 4x² - 6x
Thus, the polynomial function in standard form is:
f(x) = 2x³ - 4x² - 6x
Now that we have the polynomial function in standard form, f(x) = 2x³ - 4x² - 6x, we can compare it with the options provided in the problem statement. The options are:
- x³ - 2x² - 3x
- x² - 2x - 3
- 2x² - 4x - 6
- 2x³ - 4x² - 6x
By direct comparison, we can see that the polynomial we derived, 2x³ - 4x² - 6x, matches option 4 exactly. The other options do not match the degree or the coefficients of our derived polynomial. Option 1 has the correct terms but is missing the factor of 2. Options 2 and 3 are quadratic functions, while our polynomial is cubic, making them incorrect choices.
In this article, we successfully identified the polynomial function that has x-intercepts of 3, 0, and -1 and passes through the point (1, -8). By constructing the polynomial in factored form using the x-intercepts, we then used the given point to solve for the constant coefficient. Expanding the polynomial into standard form allowed us to directly compare it with the provided options. The correct function is f(x) = 2x³ - 4x² - 6x. This process demonstrates a powerful technique for determining polynomial functions from their key features, a skill that is invaluable in numerous mathematical and scientific applications. Understanding how to connect roots, points, and polynomial forms empowers us to model and analyze various real-world phenomena with greater precision and insight. This exploration not only reinforces our understanding of polynomial functions but also highlights their versatility and importance in problem-solving.
Q: What are x-intercepts and why are they important in defining a polynomial function?
X-intercepts, also known as roots or zeros, are the points where the graph of a function intersects the x-axis. These points are crucial in defining a polynomial function because they directly relate to the factors of the polynomial. If a polynomial has an x-intercept at x = a, then (x - a) is a factor of the polynomial. Knowing the x-intercepts allows us to construct the factored form of the polynomial, which is a fundamental step in identifying the function.
Q: How does the point (1, -8) help in determining the specific polynomial function?
The point (1, -8) provides a specific coordinate that the graph of the polynomial function must pass through. This information is crucial for finding the constant coefficient, k, in the factored form of the polynomial. By substituting x = 1 and f(x) = -8 into the equation f(x) = k(x - a)(x - b)(x - c), where a, b, and c are the x-intercepts, we can solve for k. This constant scales the polynomial vertically and ensures that the graph passes through the given point, thus uniquely defining the polynomial.
Q: Can there be multiple polynomials that satisfy the given x-intercepts?
Yes, there can be infinitely many polynomials that satisfy the given x-intercepts if we don't have additional information. The x-intercepts define the roots of the polynomial, but they don't uniquely determine the polynomial itself. The polynomials could have different vertical stretches, compressions, or reflections, which are controlled by the constant coefficient k. Without a specific point that the polynomial passes through, we can only define a family of polynomials with the same roots. The additional point provides the constraint needed to find a unique solution.
Q: Why is expanding the polynomial necessary to solve the problem?
Expanding the polynomial from its factored form to its standard form (f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀) is necessary for direct comparison with the given options. The standard form makes it easier to identify the degree of the polynomial and the coefficients of each term. This direct comparison allows us to determine which of the provided functions matches the one we constructed using the x-intercepts and the given point. While the factored form is useful for understanding the roots, the standard form is essential for algebraic manipulation and comparison.
Q: What is the significance of the degree of the polynomial in this context?
The degree of the polynomial indicates the highest power of x in the function. In this context, the x-intercepts 3, 0, and -1 suggest that we are dealing with a cubic polynomial (degree 3) because there are three distinct roots. The degree of the polynomial is significant because it determines the general shape of the graph and the maximum number of turning points. In comparing our derived polynomial with the given options, the degree helps us quickly eliminate options that are not cubic functions (e.g., quadratic functions).
In this article, we've covered a lot of ground on polynomial functions. Our primary focus was on x-intercepts and their crucial role. Understanding x-intercepts is fundamental. We explored how x-intercepts, combined with an additional point, help define a unique polynomial. Polynomial functions are expressions with variables raised to non-negative integer powers. The x-intercepts (roots) are where the function crosses the x-axis. The factored form of a polynomial relies heavily on its x-intercepts. A point, like (1, -8), helps determine the vertical stretch or compression of the polynomial. Finding the polynomial involves using x-intercepts to create factors. Expanding helps match the polynomial to standard forms. The degree of a polynomial function tells us its basic shape and maximum turning points. Identifying a polynomial with x-intercepts and a point is a core skill.
