Find Polynomial Function With X-Intercepts -1, 0, And 2 And Passing Through (1, -6)

Finding the right polynomial function that satisfies specific conditions can be a fascinating challenge in mathematics. In this article, we will delve into a step-by-step approach to identify the polynomial function with given x-intercepts and a point it passes through. Understanding polynomial functions is crucial in various fields, including engineering, physics, and computer science. This article aims to provide a comprehensive guide to solving such problems, ensuring clarity and depth in our explanation.

Understanding Polynomial Functions and X-Intercepts

Polynomial functions are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They play a vital role in modeling real-world phenomena and solving mathematical problems. The general form of a polynomial function is:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$

where $a_n, a_{n-1}, ..., a_1, a_0$ are coefficients and $n$ is a non-negative integer representing the degree of the polynomial.

The x-intercepts of a polynomial function are the points where the graph of the function crosses the x-axis. These are the values of $x$ for which $f(x) = 0$. Knowing the x-intercepts can significantly simplify the process of finding the polynomial function, as they provide the roots of the polynomial. For instance, if a polynomial has x-intercepts at $x = -1$, $x = 0$, and $x = 2$, this means that $f(-1) = 0$, $f(0) = 0$, and $f(2) = 0$. These roots directly translate into factors of the polynomial, which is a fundamental concept we will utilize in solving our problem.

The relationship between roots and factors is a cornerstone of polynomial algebra. If $x = a$ is a root of a polynomial, then $(x - a)$ is a factor of the polynomial. This principle allows us to construct a polynomial function from its roots. In our case, with x-intercepts at $-1$, $0$, and $2$, we know that $(x + 1)$, $(x)$, and $(x - 2)$ are factors of the polynomial. By multiplying these factors, we can create a basic form of the polynomial function. However, this basic form may need to be adjusted to satisfy other given conditions, such as passing through a specific point.

Constructing a Polynomial Function from X-Intercepts

To begin, we use the given x-intercepts $-1, 0,$ and $2$ to construct a preliminary form of the polynomial function. Since these are the x-intercepts, the function will have the factors $(x - (-1))$, $(x - 0)$, and $(x - 2)$, which simplify to $(x + 1)$, $(x)$, and $(x - 2)$.

Therefore, we can write a polynomial function in the form:

$$f(x) = k imes x imes (x + 1) imes (x - 2)$$

where $k$ is a constant factor. This constant factor is crucial because there are infinitely many polynomial functions with the same x-intercepts, differing only by a vertical stretch or compression. To determine the exact function, we need additional information, which is provided in the form of a point that the polynomial passes through.

The next step involves expanding this factored form to get a standard polynomial expression. Multiplying the factors, we get:

$$f(x) = k imes x imes (x^2 - 2x + x - 2)$$

$$f(x) = k imes x imes (x^2 - x - 2)$$

$$f(x) = k(x^3 - x^2 - 2x)$$

This expression represents a family of polynomial functions that all share the same x-intercepts. The value of $k$ determines the specific function within this family that satisfies our additional condition: passing through the point $(1, -6)$. This point will allow us to solve for $k$ and thus determine the unique polynomial function we are seeking.

Using the Point (1, -6) to Determine the Constant Factor

The given point $(1, -6)$ provides the necessary information to find the constant factor $k$. This point tells us that when $x = 1$, the function value $f(x)$ is $-6$. We can use this information by substituting these values into our polynomial function:

$$f(1) = k(1^3 - 1^2 - 2(1)) = -6$$

This substitution creates an equation that we can solve for $k$. By simplifying the expression inside the parentheses, we get:

$$k(1 - 1 - 2) = -6$$

$$k(-2) = -6$$

Now, we can solve for $k$ by dividing both sides of the equation by $-2$:

k = rac{-6}{-2} = 3

So, the constant factor $k$ is equal to $3$. This means that the specific polynomial function we are looking for is three times the basic polynomial function we derived from the x-intercepts. This scaling factor ensures that the polynomial passes through the point $(1, -6)$, fulfilling all the given conditions.

Knowing the value of $k$, we can now substitute it back into the polynomial function to obtain the final expression. This completes the process of constructing the polynomial function that satisfies both the x-intercept conditions and the point condition.

Final Polynomial Function

Now that we have determined the value of $k$ to be $3$, we can substitute it back into the polynomial function:

$$f(x) = k(x^3 - x^2 - 2x)$$

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

Expanding this, we get:

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

This is the polynomial function that has x-intercepts at $-1$, $0$, and $2$, and passes through the point $(1, -6)$. We can verify this by plugging in the x-intercepts and the point $(1, -6)$ into the function to ensure they satisfy the equation. When $x = -1$, $x = 0$, and $x = 2$, the function should equal zero, and when $x = 1$, the function should equal $-6$.

To verify, let's check the point $(1, -6)$:

$$f(1) = 3(1)^3 - 3(1)^2 - 6(1) = 3 - 3 - 6 = -6$$

This confirms that our polynomial function indeed passes through the point $(1, -6)$. The process of finding this function involved several key steps: identifying the factors from the x-intercepts, constructing a general polynomial form, determining the constant factor using the given point, and finally, writing the complete polynomial function. This method can be applied to similar problems involving polynomial functions and their properties.

Comparing with the Given Options

Now that we have found the polynomial function, we can compare it with the given options to select the correct answer. The polynomial function we derived is:

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

Let's compare this with the options provided:

A. $f(x) = x^3 - x^2 - 2x$ B. $f(x) = 3x^3 - 3x^2 - 6x$ C. $f(x) = x^3 + x^2 - 2x$ D. $f(x) = 3x^3 + 3x^2 - 6x$

By direct comparison, we can see that option B matches our derived polynomial function exactly. Therefore, option B is the correct answer.

Options A, C, and D do not match our derived polynomial. Option A lacks the correct constant factor, while options C and D have different signs in their terms. This highlights the importance of accurately determining the constant factor and ensuring the correct signs in the polynomial expression. The ability to compare and verify our result with the given options is a crucial step in problem-solving, ensuring that we have arrived at the correct solution.

Conclusion

In conclusion, we have successfully identified the polynomial function that has x-intercepts $-1$, $0$, and $2$ and passes through the point $(1, -6)$. The process involved understanding the relationship between x-intercepts and factors of a polynomial, constructing a general polynomial form, using the given point to determine the constant factor, and comparing our result with the provided options. The correct polynomial function is:

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

This problem demonstrates a fundamental concept in polynomial algebra, which is the ability to construct a polynomial function from its roots and a point it passes through. This skill is essential in various mathematical applications and provides a solid foundation for more advanced topics in algebra and calculus. Understanding the steps involved and practicing similar problems will enhance your problem-solving abilities and deepen your understanding of polynomial functions.

By following this step-by-step approach, you can confidently tackle similar problems involving polynomial functions and their properties. Remember to always verify your solution to ensure accuracy and a thorough understanding of the concepts involved.