Which is the reminder resulted when P is divided by Q? P=(x-2)^2010 + x-2 Q=(x-3)
P= (x-2)^2010 + x -2
==> P= Q*f + R Where, f is a function and R is the remainder.
To calculate the remainder , we will substitute with Q's root (3)
P(3) = Q(3) f(3) + R
1+3-2 = R
P(x) = (x-2)^2010+x-2. Q (x) = (x-3).
To find the remainder, R when P(x) is divided by Q(x)/
Let P(x)/Q(x) = Quotent K(x) and remaniderR(x).
So, P(x) = Q(x)*K(x) +R, where R a constan R has to be at least one degree less than Q(x) which is x-3.
So (x-2)^2010 +(x-2) = (x-3)K(x) +R. Put x=3 and we get:
(3-2)^2010 +(3-2) = (3-3)K(3) +R.Or
1+1 = 0 +R. Or R = 2.
First, we'll remember the rule of division with reminder.
According to the rule, the degree of the reminder has to be smaller than the degree of Q. In our case, Q is a linear function, so the reminder will be a constant.
First, let's find out the roots of Q.
Now, we'll substitute the root of Q, into the rule of division with reminder:
f(3)=Q(3)*Q1+a, where f(3)=(3-2)^2010 + 3-2 = 1+1=2 and Q(3)=0
The reminder is R=2