A Polynomial P(x) has remainder 5 when divided by (x-1) and has remainder 8 when divided by (x+2).
When divided by (x-1)(x+2), what is the remainder?
Much appreciated for any help given.
If P(x) is divided by (x-1)(x+2), the remainder will have the form Ax + B. Because the remainder must have a lower power than the thing you are dividing by, which is a quadratic. So the remainder is at most linear.
In other words, P(x) = (x-1)(x+2)Q(x) + Ax + B.
where Q(x) is some unknown function which is two powers lower than P(x). We don't really need to know anything more about Q(x).
At this point, we can use the info given. Since P(x) has remainder 5 when divided by (x−1), and remainder 8 when divided by (x+2), the Remainder Theorem tells us that P(1) = 5 and P(−2) = 8.
Now we can solve for A and B.