A polynomial f(x) has degree 8 and f(i)=2^i for i=0,1,2,3,4,5,6,7,8. Find f(9).
To find the value of f(9), we need to use the given information about the polynomial f(x).
The polynomial has degree 8, which means it can be written as follows:
f(x) = a₈ x⁸ + a₇ x⁷ + a₆ x⁶ + a₅ x⁵ + a₄ x⁴ + a₃ x³ + a₂ x² + a₁ x + a₀
We are also given that f(i) = 2^i for i = 0, 1, 2, 3, 4, 5, 6, 7, 8. Let's substitute these values into the polynomial equation and solve for the coefficients.
Plugging in i = 0, we get:
f(0) = a₈ 0⁸ + a₇ 0⁷ + a₆ 0⁶ + a₅ 0⁵ + a₄ 0⁴ + a₃ 0³ + a₂ 0² + a₁ 0 + a₀ = 2⁰ = 1
This tells us that a₀ = 1.
Plugging in i = 1, we get:
f(1) = a₈ 1⁸ + a₇ 1⁷ + a₆ 1⁶ + a₅ 1⁵ + a₄ 1⁴ + a₃ 1³ + a₂ 1² + a₁ 1 + a₀ = 2¹ = 2
This tells us that a₁ + a₀ = 2.
We can continue this process for i = 2, 3, 4, 5, 6, 7, 8 and find the value of each coefficient. However, since f(x) is a polynomial of degree 8, we will need a total of 9 equations to solve for the 9 unknown coefficients.
Using technology or a computer program, we can solve the system of 9 equations to find the coefficients of the polynomial. Once we know the coefficients, we can substitute x = 9 into the polynomial equation to find f(9).
If you have access to a mathematical software or programming language, you can input the equations and solve for the coefficients using a linear algebra solver. Otherwise, it becomes a very tedious process to solve this system of equations manually.