Can you give me some help with question 10 in the Polynomials section of PWN (pg 149)? I plug in x=-3, per your suggestion, and I get 41 as the answer on the left-hand side of the equation and 3(-3) – 13 + a on the right. That means 41 = -22 + a, or a = 63. What am I missing? Thanks!
Sure! Here’s the question:
The key to this is recognizing that the original equation you’re getting is in quotient-remainder form: is the quotient with a remainder of . Once you recognize that, you have a few options. You can actually do the division to find the remainder, you can multiply the whole equation by and solve for , or you can use the remainder shortcut (since you’re only asked for the remainder in this question) by plugging into only the original polynomial.
The shortcut: When a polynomial is divided by , where is a constant, you can find the remainder simply by finding . In this case, that means we can find the remainder when is divided by simply by simplifying , which as you note comes out to 41.
Note that we actually can’t plug in in the original equation: we’d get division by zero on both sides.
I’m not going to do the whole long division here, but because I know that shortcut is confusing, I want to stress that just solving the equation for is also a decent way to go. To begin, multiply everything by .