Use the remainder theorem to find the remainder when P(x) = x^3-2ax^2+1 is divided by x-a-1.
x - a - 1 = x - (a+1)
The remainder will be P(a+1):
P(a+1) = (a+1) ³ -2a(a+1) ² +1
Expand and simplify.
To find the remainder when P(x) is divided by x-a-1 using the remainder theorem, we can substitute x = a+1 into the polynomial P(x) and compute the result.
Step 1: Substitute x = a+1 into P(x).
P(a+1) = (a+1)^3 - 2a(a+1)^2 + 1
Step 2: Simplify the expression.
P(a+1) = (a+1)(a+1)(a+1) - 2a(a+1)(a+1) + 1
P(a+1) = (a+1)(a+1)(a+1) - 2a(a+1)(a+1) + 1
P(a+1) = (a+1)^3 - 2a(a+1)^2 + 1
Step 3: Expand and simplify further.
P(a+1) = (a+1)(a^2 + 2a +1) - 2a(a^2 + 2a + 1) + 1
P(a+1) = (a+1)(a^2 + 2a + 1) - 2a^3 - 4a^2 - 2a + 1
P(a+1) = a^3 + 2a^2 + a + a^2 + 2a + 1 - 2a^3 - 4a^2 - 2a + 1
P(a+1) = -a^3 -a^2 + 2a
Step 4: The remainder when P(x) is divided by x-a-1 is -a^3 -a^2 + 2a.
Therefore, the remainder is -a^3 -a^2 + 2a when P(x) is divided by x-a-1.