Introduction: This article provides a step-by-step guide on constructing a polynomial function of degree 3 with given zeros and a specific value at a certain point. It elucidates the process through algebraic manipulation and the use of an online graphing calculator. By understanding these steps, you can construct similar polynomial functions with ease.

How to Construct a Polynomial Function of Degree 3 with Given Zeros and Value

To begin, consider a polynomial function ?x of degree 3 with given zeros 2, -3, and 5, and a value ?3 6. We aim to construct a function that satisfies these conditions.

Step 1: Write the General Form of the Polynomial Function

Given the zeros 2, -3, and 5, the polynomial function ?x can be written in the form:

?x a(x - 2)(x 3)(x - 5)

Here, a is a scaling factor that will be determined later.

Step 2: Impose the Given Value ?3 6

We know that ?3 6, so we need to substitute x 3 into the polynomial and set it equal to 6:

a * 3 - 2 * 3 3 * 3 - 5 6

Substituting these values into the equation:

a(3 - 2)(3 3)(3 - 5) 6

a * 1 * 6 * (-2) 6

-12a 6

Solving for a gives:

a -1/2

Step 3: Substitute a and Write the Polynomial Function

Using the value of a -1/2, we can write the polynomial function as:

?x -1/2(x - 2)(x 3)(x - 5)

Expanding this, we get:

?x -1/2(x3 - 2x2 - 11x 30)

?x -1/2x3 x2 11/2x - 15

Using the Online Graphing Calculator

To verify this polynomial, you can graph it using an online graphing calculator like Desmos. Here are the steps:

Enter the function as ?(x) -1/2(x - 2)(x 3)(x - 5). Identify the three given zeros and their values. Check if the function passes through the point (3, 6).

Vieta's Rule and Additional Observations

Using Vieta’s rule, we can confirm the relationships between the coefficients of the polynomial and its zeros. For a polynomial of the form ax3 bx2 cx d, the relationships are:

2 - 3 - 5 -b/a 2 * -3 3 * -5 -3 * 5 c/a 2 * -3 * 5 -d/a

Substituting the known values, we get:

-6 -b/a rarr; b 6a -6a - 15a - 15 c/a rarr; c -21a -30a -d/a rarr; d 30a

Substituting into the expanded form:

?x -1/2x3 6ax2 - 21ax 30a

?x -1/2x3 - 6x2 - 11x 15

Summary

In conclusion, constructing a polynomial function of degree 3 with given zeros and a specific value at a certain point involves following a systematic approach. This article has detailed the steps required to determine the function ?x -1/2(x - 2)(x 3)(x - 5), which satisfies the given conditions. The use of an online graphing calculator can further verify the accuracy of the constructed function.