Find Polynomial Function With Given Intercepts And Point

Hey guys! Today, we're diving into the world of polynomial functions. Specifically, we're going to tackle a common problem: finding the equation of a polynomial function when we know its x-intercepts and a point it passes through. This is a super useful skill in algebra and calculus, so let's break it down step-by-step.

Understanding the Basics of Polynomial Functions

Before we jump into the problem, let's quickly review what polynomial functions are all about. At its core, a polynomial function is a mathematical expression consisting of variables and coefficients, combined using only addition, subtraction, and non-negative integer exponents. Think of it like this: it's a smooth, continuous curve that can wiggle and turn, but it doesn't have any sharp corners or breaks. Key components of understanding polynomial functions are the x-intercepts, also known as roots or zeros, which are the points where the function crosses the x-axis. These intercepts are crucial because they directly relate to the factors of the polynomial. For instance, if a polynomial has an x-intercept at x = a, then (x - a) is a factor of the polynomial. This relationship forms the foundation for constructing the polynomial function when the intercepts are known. The degree of the polynomial, which is the highest power of the variable, dictates the maximum number of roots the polynomial can have and influences the overall shape of its graph. Understanding the relationship between the roots, factors, and degree is essential for solving problems related to polynomial functions. Moreover, the leading coefficient plays a significant role in determining the end behavior of the polynomial, indicating how the function behaves as x approaches positive or negative infinity. This interplay between coefficients, roots, and the degree of the polynomial makes polynomial functions versatile tools for modeling various real-world phenomena, from physics to economics.

Understanding polynomial functions is critical for solving many mathematical problems. These functions are characterized by their smooth, continuous curves and can be expressed in the general form: f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are constants and n is a non-negative integer. The degree of the polynomial is the highest power of x, which in this case is n. The x-intercepts are the points where the graph of the function crosses the x-axis, meaning f(x) = 0. These x-intercepts are also known as the roots or zeros of the polynomial. Each root corresponds to a factor of the polynomial. For instance, if x = a is a root, then (x - a) is a factor. This is a fundamental concept we’ll use to build our polynomial function. The number of roots a polynomial can have is limited by its degree. A polynomial of degree n can have at most n roots, though some roots may be repeated. For example, a cubic polynomial (degree 3) can have up to three roots. The coefficients of the polynomial influence the shape and position of the graph. The leading coefficient (a_n) is particularly important as it determines the end behavior of the function – whether the graph rises or falls as x approaches positive or negative infinity. In addition to intercepts, another crucial aspect of polynomial functions is their smooth, continuous nature. Unlike functions with sharp corners or breaks, polynomials flow smoothly across their entire domain. This property makes them useful in modeling many real-world phenomena, such as curves in physics or economic trends. Understanding how to manipulate and analyze polynomial functions is a cornerstone of both algebra and calculus.

Setting Up the Problem: Intercepts and a Point

Now, let's dive into the problem at hand. We're given the x-intercepts of a polynomial function: -1, 0, and 2. Remember, x-intercepts are the points where the graph of the function crosses the x-axis, meaning the function's value (y) is zero at these points. We're also given that the function passes through the point (1, -6). This is extra information that will help us nail down the exact polynomial function. The x-intercepts of a polynomial function are critical pieces of information because they directly tell us the roots of the polynomial equation. Specifically, the roots are the x-values for which the polynomial function equals zero, where the graph intersects the x-axis. Each root corresponds to a factor of the polynomial, and this correspondence is a cornerstone for building the polynomial equation. When we know that the x-intercepts are -1, 0, and 2, it informs us immediately that x = -1, x = 0, and x = 2 are the roots of the polynomial. This means the factors of our polynomial will be of the form (x - root). Thus, the factors corresponding to our given intercepts are (x - (-1)), (x - 0), and (x - 2), which simplify to (x + 1), x, and (x - 2). These factors form the foundation upon which we will construct the polynomial function. By multiplying these factors together, we obtain a polynomial that has the correct x-intercepts but may not yet satisfy the condition of passing through the given point (1, -6). This is where the additional point becomes crucial, allowing us to determine the precise scale or coefficient of the polynomial, thereby uniquely defining the polynomial function that meets all the given criteria.

Building the General Form of the Polynomial

Since we know the x-intercepts, we can write the general form of the polynomial. If a polynomial has x-intercepts at -1, 0, and 2, it means it has factors of (x + 1), x, and (x - 2). So, our polynomial can be written as:

f(x) = a * x * (x + 1) * (x - 2)

where 'a' is a constant we need to determine. This constant will scale the polynomial, ensuring it passes through the point (1, -6). Now, let's elaborate on why this general form works. The factors (x + 1), x, and (x - 2) directly arise from the given x-intercepts of the polynomial function, -1, 0, and 2. Each x-intercept represents a root of the polynomial equation f(x) = 0. According to the factor theorem, if r is a root of a polynomial, then (x - r) is a factor of the polynomial. Thus, each x-intercept corresponds to a linear factor. Specifically, an x-intercept of -1 gives us the factor (x - (-1))*, which simplifies to (x + 1). Similarly, an x-intercept of 0 gives the factor x, and an x-intercept of 2 gives the factor (x - 2). Multiplying these factors together creates a polynomial that has roots at the required x-intercepts. However, this product alone does not uniquely define the polynomial function because any constant multiple of this product will also have the same roots. That’s where the constant 'a' comes in. The constant 'a' acts as a scaling factor, which can vertically stretch or compress the polynomial. By multiplying the product of the factors by 'a', we account for all polynomials that have the given roots. This general form captures an entire family of polynomials sharing the same x-intercepts, and the specific value of 'a' is determined by additional information, such as a point the polynomial must pass through.

Finding the Constant 'a'

This is where the point (1, -6) comes in handy. We know that when x = 1, f(x) = -6. Let's plug these values into our general form:

-6 = a * 1 * (1 + 1) * (1 - 2)

Now, we simplify:

-6 = a * 1 * 2 * (-1) -6 = -2a

Divide both sides by -2:

a = 3

So, we've found our constant! The constant 'a' plays a crucial role in determining the vertical stretch or compression of the polynomial function. Without this constant, we would have a general shape that crosses the x-axis at the correct points, but we wouldn't be able to ensure that the function passes through the specific point (1, -6). By substituting the coordinates of this point into our general form of the polynomial, we create an equation that allows us to solve for 'a'. In this case, the equation -6 = a * 1 * (1 + 1) * (1 - 2) arises from the fact that when x = 1, the function's value, f(x), is -6. The equation simplifies to -6 = -2a, which is a straightforward linear equation. Solving for 'a' involves isolating 'a' on one side of the equation, which we achieve by dividing both sides by -2. This yields a = 3, meaning that the vertical stretch factor of the polynomial is 3. This specific value ensures that our polynomial not only has the correct x-intercepts but also precisely aligns with the given point (1, -6). Understanding the role of 'a' in scaling the polynomial is essential for accurately defining the function that meets all the specified conditions.

Writing the Final Polynomial Function

Now that we know a = 3, we can substitute it back into our general form:

f(x) = 3 * x * (x + 1) * (x - 2)

Let's expand this to get the polynomial in standard form:

f(x) = 3x * (x^2 - 2x + x - 2) f(x) = 3x * (x^2 - x - 2) f(x) = 3x^3 - 3x^2 - 6x

So, the polynomial function is f(x) = 3x^3 - 3x^2 - 6x. Expanding the polynomial function into its standard form serves several key purposes. Firstly, it provides a clearer representation of the polynomial's structure and degree. The standard form of a polynomial is expressed as a sum of terms, each consisting of a coefficient and a power of x. This form makes it easy to identify the degree of the polynomial, which is the highest power of x, and the coefficients of each term. In our case, expanding f(x) = 3x * (x + 1) * (x - 2) involves first multiplying the linear factors (x + 1) and (x - 2), which yields a quadratic expression. Then, we multiply the result by 3x. The step-by-step expansion ensures that we correctly distribute each term and combine like terms. This process is crucial for avoiding errors and obtaining the correct standard form. The final expanded form, f(x) = 3x³ - 3x² - 6x, is a cubic polynomial (degree 3) with a leading coefficient of 3. This standard form allows for easier analysis of the polynomial’s behavior, such as its end behavior and the shape of its graph. It also facilitates comparison with other polynomials and makes it straightforward to apply various polynomial operations, like differentiation and integration in calculus. Overall, expanding the polynomial to its standard form provides a more accessible and practical representation for further mathematical analysis and applications.

Solution

The polynomial function that has x-intercepts -1, 0, and 2 and passes through the point (1, -6) is:

f(x) = 3x^3 - 3x^2 - 6x

Conclusion

There you have it! We successfully found the polynomial function by using the x-intercepts to build the general form and then using the given point to solve for the constant. This method is a powerful tool for working with polynomial functions. Keep practicing, and you'll become a polynomial pro in no time!

Keywords: Polynomial function, x-intercepts, roots, factors, degree, leading coefficient, general form, constant, standard form.

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