Taylor's Theorem and Polynomial Approximation

Taylor's theorem is a mathematical formula that allows us to approximate a function using a polynomial. It provides a way to express a function as an infinite series of terms involving the function's derivatives at a specific point.

Expressing a Polynomial in Powers of (x-2)

To express the polynomial 2x^3 + 7x^2 + x - 6 in powers of (x - 2), we can use Taylor's theorem. The general form of Taylor's theorem is as follows:

f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...

In this case, we want to express the polynomial in powers of (x - 2), which means we want to find values for a, f(a), f'(a), f''(a), f'''(a), and so on.

Step 1: Choose a Value for a

We can choose a = 2 since we want to express the polynomial in powers of (x - 2).

Step 2: Find the Values of f(a), f'(a), f''(a), f'''(a), and so on

To find these values, we need to calculate the function and its derivatives at x = 2.

f(x) = 2x^3 + 7x^2 + x - 6

f(2) = 2(2)^3 + 7(2)^2 + 2 - 6 = 30

f'(x) = 6x^2 + 14x + 1

f'(2) = 6(2)^2 + 14(2) + 1 = 49

f''(x) = 12x + 14

f''(2) = 12(2) + 14 = 38

f'''(x) = 12

f'''(2) = 12

Step 3: Apply Taylor's Theorem

Using the values we calculated, we can now apply Taylor's theorem to express the polynomial in powers of (x - 2).

f(x) = f(2) + f'(2)(x - 2) + f''(2)(x - 2)^2/2! + f'''(2)(x - 2)^3/3! + ...

f(x) = 30 + 49(x - 2) + 38(x - 2)^2/2! + 12(x - 2)^3/3! + ...

Simplifying the expression further will provide the polynomial expressed in powers of (x - 2).