An unknown polynomial f(x) of degree 37 yields a remainder of 1 when divided by x – 1, a remainder of 3 when divided by x – 3, a remainder of 21 when divided by x – 5.

Question

Find the remainder when f(x) is divided by (x – 1)(x – 3)(x – 5).

Answer 1

If you divide by a cubic polynomial, the remainder will be a quadratic, r(x) = ax^2+bx+c

Now, you know that
r(1) = 1
r(3) = 3
r(5) = 21

a+b+c = 1
9a+3b+c = 3
25a+5b+c = 21

r(x) = 2x^2-7x+6