The Remainder Theorem states that if a polynomial f(x) is divided by x - a, then the remainder of the division is equal to f(a). In other words, if we divide a polynomial by x - a, the value of the polynomial at x = a is equal to the remainder.
Using the Remainder Theorem, we can find P(a) for the polynomial f(x) = x^3 - 2x^2 - 3x + 6 when a = 2:
First, we need to create a polynomial that is equivalent to f(x) when divided by x - a. This is done by performing synthetic division, using a as the divisor:
2 | 1 -2 -3 6
2 0 -6
1 0 -3 0
The result of the synthetic division is the polynomial 1x^2 + 0x - 3 with a remainder of 0. This means that f(x) can be expressed as:
f(x) = (x - 2)(x^2 + 0x - 3)
We can now use the Remainder Theorem to find P(a) for f(x) when a = 2. This means that we need to evaluate the polynomial x^2 + 0x - 3 at x = 2:
P(2) = 2^2 + 0(2) - 3 = 1
Therefore, P(2) = 1.
The result P(2) means that when we divide the polynomial f(x) = x^3 - 2x^2 - 3x + 6 by x - 2, the remainder is 1.