The question: The remainder obtained when 3x^4 + 7x^3 + 8x^2 -2x -3 is divided by x + 1 is?
The answer is 3. My book says to plug in -1 for x (which results with 3). Why should I do this?
Because that's the way you find a remainder.
For example if you want to solve the equation $\displaystyle f(x)=3x^4+7x^3+8x^2-2x-3=0$, in first instance you're going to search for a number $\displaystyle x_1$ wherefore $\displaystyle f(x_1)=0$, that means if you divide $\displaystyle f(x)$ by $\displaystyle (x-x_1)$ you'll obtain remainder 0, in this case if you enter -1 you see $\displaystyle f(x)=3$ so the remainder is 3. To have a closer look you can make the Horner scheme.
Hmm, thanks. Makes a little more sense. I guess I still need to work a few hands-on examples to make sense of why we do that. Let me ask a few questions to get a more thorough understanding:
1.) Why do we set the first equation equal to 0?
2.) Why does dividing by the the x-x1 give us remainder 0?
That was just an example (another case), and because $\displaystyle x_1$ is a zero therefore if you divive $\displaystyle f(x)$ by $\displaystyle (x-x_1)$ you'll obtain remainder 0 (this is a theoretical look to it).