To determine a polynomial function $f(x)$ that fits the points given, we start by expressing $f(x)$ as a general polynomial:

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

where $n$ is the degree of the polynomial. Since there are four points, we expect the polynomial to be of degree $n \leq 3$, as a degree-3 polynomial can uniquely fit four points.

The points are: $f(1) = 12, \quad f(26) = 5, \quad f(54) = 2, \quad f(85) = 1.$

Step 1: General Form

Assume a cubic polynomial: $f(x) = ax^3 + bx^2 + cx + d$

Substitute the given points into the equation to create a system of four equations:

1. $f(1) = 12$: $a(1)^3 + b(1)^2 + c(1) + d = 12 \quad \Rightarrow \quad a + b + c + d = 12$

2. $f(26) = 5$: $a(26)^3 + b(26)^2 + c(26) + d = 5 \quad \Rightarrow \quad 17576a + 676b + 26c + d = 5$

3. $f(54) = 2$: $a(54)^3 + b(54)^2 + c(54) + d = 2 \quad \Rightarrow \quad 157464a + 2916b + 54c + d = 2$

4. $f(85) = 1$: $a(85)^3 + b(85)^2 + c(85) + d = 1 \quad \Rightarrow \quad 614125a + 7225b + 85c + d = 1$

Step 2: Solve the System of Equations

The system of four equations is:

1. $a + b + c + d = 12$
2. $17576a + 676b + 26c + d = 5$
3. $157464a + 2916b + 54c + d = 2$
4. $614125a + 7225b + 85c + d = 1$

We solve this system (numerically or symbolically) to determine $a$, $b$, $c$, and $d$.

Let me compute the solution. sympy import symbols, Eq, solve

Define variables

a, b, c, d = symbols('a b c d')

Define the equations based on the given points

eq1 = Eq(a + b + c + d, 12) eq2 = Eq(17576a + 676b + 26c + d, 5) eq3 = Eq(157464a + 2916b + 54c + d, 2) eq4 = Eq(614125a + 7225b + 85*c + d, 1)

Solve the system of equations

solution = solve([eq1, eq2, eq3, eq4], (a, b, c, d)) solutionThe coefficients of the polynomial $f(x) = ax^3 + bx^2 + cx + d$ are as follows:

$a = -\frac{1207}{50891925}, \quad b = \frac{70333}{13571180}, \quad c = -\frac{82089737}{203567700}, \quad d = \frac{84128399}{6785590}.$

Final Polynomial:

Substitute these coefficients back into $f(x)$: $f(x) = -\frac{1207}{50891925}x^3 + \frac{70333}{13571180}x^2 - \frac{82089737}{203567700}x + \frac{84128399}{6785590}.$

Would you like a simplified or approximate form of this polynomial, or do you have further questions?

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Related Questions:

1. How do you verify that the polynomial fits all given points?
2. What is the significance of the degree of the polynomial in interpolation?
3. Could this polynomial be expressed in a simpler or alternative form?
4. How does the Vandermonde matrix help in solving such systems?
5. Can this approach be extended to higher-degree polynomials?

Tip:

When solving such systems, numerical methods or software tools like Python or MATLAB can save significant time!