A question on polynomials practice question #10 (p. 149 in PWN the SAT Math Guide): I understand how to solve using polynomial long division but can you explain your shortcut from the answer explanation? How would we know to plug in -3?

Sure, here’s the question:

The shortcut in the solution is to plug in , and that comes from the bolded text on page 145 of the book. If you want to know the remainder when a polynomial is divided by , where is some constant, then plug into the polynomial for and see what comes out. If you get 0, then is a factor of the polynomial. If you get something other than 0, it’s the remainder.

Since this question is just asking for , the remainder, we can just follow that shortcut. Dividing by is the same as dividing by , which is why we plug in . Does that help?

There’s more on this concept on pages 144 and 145 of the book.