A polynomial P(x) has remainder R x
+S on division by (x-a) (x-b). If a is not
equal to b, express the constants R
and S in terms of a, b, P(a), P(b). Find
the remainder when P(x) is divided
by (x-a)^2.
To find the constants R and S, we need to use the Remainder Theorem, which states that if a polynomial P(x) is divided by (x-a) and (x-b), the remainders will be equal to P(a) and P(b), respectively.
Let's start with the remainder when P(x) is divided by (x-a). Since the remainder is Rx + S, we have:
P(a) = Ra + S .....(1)
Next, let's consider the remainder when P(x) is divided by (x-b). Again, the remainder is Rx + S, so we have:
P(b) = Rb + S .....(2)
We now have a system of two equations with two unknowns (R and S). We can solve this system to find the values of R and S.
First, subtract equation (2) from equation (1) to eliminate S:
P(a) - P(b) = Ra - Rb
Now, let's solve for R:
R = (P(a) - P(b)) / (a - b)
Substitute the value of R in equation (1) to find S:
P(a) = (P(a) - P(b))(a) / (a - b) + S
P(a)(a - b) = (P(a) - P(b))(a) + S(a - b)
(P(a) - P(b))(a - b) = S(a - b)
Finally, we can solve for S:
S = (P(a) - P(b))(a - b) / (a - b)
S = P(a) - P(b)
So, the constants R and S in terms of a, b, P(a), and P(b) are:
R = (P(a) - P(b)) / (a - b)
S = P(a) - P(b)
To find the remainder when P(x) is divided by (x-a)^2, we can use the Synthetic Division method. Here are the steps:
1. Write the polynomial P(x) in descending order.
2. Set up the synthetic division table using (x-a) as the divisor.
3. Divide the first term (highest degree term) of P(x) by (x-a).
4. Multiply the remainder obtained in step 3 by (x-a) and subtract from the next term of P(x).
5. Repeat step 4 until all terms of P(x) are divided.
6. The last term in the resulting row will be the remainder when P(x) is divided by (x-a)^2.
By following these steps, you can find the remainder when P(x) is divided by (x-a)^2.