solve f(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0 using  (90, 114.923) (110, 102.8561) ( 130, 100.1601) (145, 100.000316)

To solve the polynomial function \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0 \) given the points \((90, 114.923)\), \((110, 102.8561)\), \((130, 100.1601)\), and \((145, 100.000316)\), we need to determine the coefficients \(a_n, a_{n-1}, \ldots, a_0\) of the polynomial that fit these points.

Let's assume the polynomial is of the form:

\[
f(x) = ax^3 + bx^2 + cx + d
\]

This is a cubic polynomial since we have four points and four unknowns.

### Step 1: Set up the system of equations

For each point \((x_i, y_i)\), substitute into the polynomial equation:

1. For \((90, 114.923)\):
   \[
   a(90)^3 + b(90)^2 + c(90) + d = 114.923
   \]

2. For \((110, 102.8561)\):
   \[
   a(110)^3 + b(110)^2 + c(110) + d = 102.8561
   \]

3. For \((130, 100.1601)\):
   \[
   a(130)^3 + b(130)^2 + c(130) + d = 100.1601
   \]

4. For \((145, 100.000316)\):
   \[
   a(145)^3 + b(145)^2 + c(145) + d = 100.000316
   \]

### Step 2: Solve the system of linear equations

The equations will be solved simultaneously to find the values of \(a\), \(b\), \(c\), and \(d\). 

Let's compute this system to find the coefficients.The coefficients of the polynomial \( f(x) = ax^3 + bx^2 + cx + d \) that fit the given points are:

\[
a = -0.000148482670995671
\]
\[
b = 0.0607129064285714
\]
\[
c = -8.27659788874459
\]
\[
d = 476.286135071429
\]

Thus, the polynomial function is:

\[
f(x) = -0.000148482670995671x^3 + 0.0607129064285714x^2 - 8.27659788874459x + 476.286135071429
\]

Would you like more details or have any questions?

Here are 8 follow-up questions:
1. How do you find the roots of this cubic polynomial?
2. Can you evaluate \( f(x) \) for a specific value of \( x \)?
3. What does the derivative of this polynomial represent?
4. How would you find the local maxima and minima of this polynomial?
5. Can you plot the graph of the polynomial function?
6. How would the function change if the coefficient \( a \) were larger?
7. How can we verify that the polynomial correctly fits the given points?
8. How would the solution process change if we had more points?

**Tip:** When dealing with polynomials, using more points than the degree of the polynomial can help identify the best-fitting function through regression.

Learn how to solve polynomial functions using regression analysis with detailed step-by-step solutions. This problem involves fitting a polynomial function to given points (90, 114.923), (110, 102.8561), (130, 100.1601), and (145, 100.000316). Explore the method of setting up a system of equations and solving for coefficients to determine the polynomial equation. Suitable for advanced high school students.

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