Determining the Polynomial Function Given Specific Intercepts and a Point

When dealing with polynomial functions, one of the tasks is to find the equation of a polynomial given its x-intercepts and a specific point through which it passes. This article provides a detailed example to illustrate the process.

Understanding the Polynomial Function

A polynomial function in one variable, say x, can be written in the general form:

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

However, when we are given x-intercepts, we can express the function in terms of its factors. If a polynomial function has roots at x -1, 0, 2, it can be represented as:

f(x) k(x 1)(x)(x - 2)

Here, k is a constant coefficient that we need to determine.

Finding the Constant Coefficient

To find the value of k, we use the information that the polynomial passes through the point (1, -6). We substitute these coordinates into the polynomial equation:

f(1) -6

Plugging in 1 for x in the factorized form:

-6 k(1 1)(1)(1 - 2)

Simplifying the right side of the equation:

-6 k(2)(1)(-1)

-6 -2k

Solving for k gives:

k 3

So, the polynomial function is:

f(x) 3(x 1)(x)(x - 2)

Expanding the Polynomial

Let#39;s expand the polynomial to get it in standard form:

f(x) 3(x 1)(x - 2)

Using the distributive property (also known as the FOIL method for binomials), we get:

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

Simplifying inside the parentheses:

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

Multiplying through by 3:

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

Verification Using a Graphing Calculator

To verify this result, we can use a graphing calculator, such as the TI-84. The steps are as follows:

Enter the function in the Y1 equation field: 3*x^3 - 3*x^2 - 6*x. Go to the table (by pressing 2nd GRAPH). Check the table for the following points: (-1, 0), (0, 0), (1, -6), and (2, 0) to confirm they lie on the curve.

Generalizing the Result

While the polynomial we found is specific to k 3, there are infinitely many higher-order polynomials that also fit the given requirements. For example:

g(x) 3x^5 - 3x^4 - 6x^3

This function also has the same roots and passes through (1, -6). It can be verified that this polynomial also has a root of multiplicity at 0.

Another polynomial with 3 real roots and 2 imaginary roots is:

h(x) 1.5x^5 - 7.5x^4 10.5x^3 - 4.5x^2 - 15x

Conclusion

By understanding the relationship between the roots of a polynomial and its form, and by using a specific point to determine the constant coefficient, we can construct the polynomial equation that meets the given criteria. Experimenting with different values of the constant can lead to an infinite number of functions that satisfy the same requirements.

Keywords: polynomial function, x-intercepts, passing through a point