an unknown polynomial f(x) of degree 32 yields a remainder of 1 when divided by x-1 and a remainder of 3 when divided by x-3, find the remainder when f(x) is divided by (x-1)(x-3).

What is the answer? and i was wondering if the part about the polynomial being degree 32 is really relevant

the remainder will always have degree less than the divisor. So, it will be of the form r(x)=(ax+b)

So, you know that
r(1)=1
r(3)=3

Now solve for a and b.

As you surmised, the degree of p(x) does not really matter.

To find the remainder when f(x) is divided by (x-1)(x-3), we can use the Remainder Theorem. According to the theorem, if a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).

Since we know the remainder when f(x) is divided by x-1 is 1, we can substitute x = 1 into f(x) and find the value of f(1). Similarly, since the remainder when f(x) is divided by x-3 is 3, we substitute x = 3 into f(x) to find the value of f(3).

To find the remainder when f(x) is divided by (x-1)(x-3), we need to find the value of f(x) that satisfies both conditions f(1) = 1 and f(3) = 3. Let's call the remainder when f(x) is divided by (x-1)(x-3) as R(x).

Since we do not have information about the specific polynomial f(x) and its degree, we cannot directly determine the remainder R(x) when f(x) is divided by (x-1)(x-3).

However, we can apply a general approach to find R(x). Let's assume that the polynomial f(x) has a degree of exactly 32. In this case, we can write f(x) as:

f(x) = Q(x) * (x-1)(x-3) + R(x),

where Q(x) is the quotient and R(x) is the remainder.

Now, the degree of the polynomial (x-1)(x-3) is 2, and we are looking for the remainder, which will have a degree less than 2.

Since f(x) is a polynomial of degree 32, the remainder R(x) must be a polynomial of degree less than 2. In other words, R(x) can be a constant or have a degree of 0.

Therefore, the remainder R(x) when f(x) is divided by (x-1)(x-3) is a constant value, which we currently do not know without additional information.

In conclusion, we cannot determine the exact remainder when f(x) is divided by (x-1)(x-3) without further information about the polynomial f(x) or additional conditions.