## mcshweezy 3 years ago What is the remainder when (3x4 + 2x3 – x2 + 2x – 14) ÷ (x + 2) ?

1. hoblos

10

2. JamesJ

@hoblos: if you answer it, you explain it!

3. mathmate

What you do is to evaluate the numerator, using x=-2 (from x+2=0), and what you get is the remainder, namely 10 as Hoblos posted.

4. cwrw238

this is called the Remainder THEorem

5. jhonyy9

(3x4+2x3-x2+2x-14) : (x+2) =3x3-4x2+7x-12 3x4+6x3 --------- 0 -4x3-x2 -4x3-8x2 ---------- 0 7x2+2x 7x2+14x ---------- 0 -12x-14 -12x-24 --------- 0 10

6. JamesJ

@cwrw: right. The idea is that given any polynomial p(x) and a linear term such as (x+2), then we write p(x) as p(x) = (x+2)q(x) + r --- (*) where q(x) is another polynomial. r is the remainder after p(x) is divided by (x+2). ===== If x=-2 is a root of the polynomial p(x), then p(-2) = 0. Hence using equation (*), 0 = p(-2) = (-2+2)q(-2) + r = 0 + r i.e., r = 0. If x=-2 is not a root of p(x), then r will not be zero. It can be calculated by evaluating p(-2), because p(-2) = (-2+2)q(-2) + r = 0 + r i.e., r = p(-2) ===== So this is why a short-cut way to finding the remainder of p(x) = 3x^4 + 2x^3 – x^2 + 2x – 14 when divided by (x+2) is just to evaluate p(x) for x = -2: r = p(-2) = 3(-2)^4 + 2(-2)^3 - (-2)^2 + 2(-2) - 14 = 48 - 16 - 4 - 4 - 14 = 10

7. malice

The answer is *not* ten, just took the test

8. mathmate

@malice If the answer is *not* ten, it could be one of the many possibilities: 1. the test has an incorrect answer 2. the test question is different from what was posted above (typo, or the test is automorphic, i.e. changes the data every time the question is displayed) 3. All three or four people who responded to the question made the same mistake with probability < \(0.01^4=10^{-8}\). Actually this is an excellent lesson to learn: We do not take answers from someone else and use it as our own work (plagiarism). Work out the problem personally so that we know HOW to do the work. Finally, you also realize that getting \(help\) or answers from this site or anywhere else for an online test is considered cheating, and is taken very seriously by the authorities.

9. Dearra13

15

10. mathmate

@dearra13 First, it is important to reproduce the question before you give the answer in case there was a mismatch somewhere. Second, at this site, it is important to show how the answer can be obtained. The answer itself is irrelevant. You can give it if you want, but the rules (as given in the code of conduct) requires you to explain how an answer can be obtained. Third, it would be instructive for you to explain how you got the answer 15, so that if everyone else has made a mistake, we can all learn from it. Conversely, if you made a mistake, I am sure everyone would be glad to chip in to identify the error, so that you can learn from it too. So please, explain how you got the answer 15.

11. Dearra13

i took the test 3 times and neither of the other answer choices are correct so it has to be 15.

12. malice

Yeah I didn't retake the test, so all I know is that it is not ten. Hopefully you're correct @dearra13! To be honest, I still do not really understand the problem, but I hope this thread helps others stuck on it! :-)

13. mathmate

@dearra13 As I mentioned earlier, if the answer is *not* ten, it could be one of the many possibilities: 1. the test has an incorrect answer 2. the test question is different from what was posted above (typo, or the test is automorphic, i.e. changes the data every time the question is displayed) 3. All three or four people who responded to the question made the same mistake. Since it is impossible to verify the exact question, we will have to leave it at that. Retaking the test does not automatically mean that the question is the same. As I said, it is more important to know how to solve the problem, than to blindly copy the answer as your own. @mcshweezy For your future career in math, I suggest you study the remainder theorem to be able to solve the problem correctly. A frequently overlooked fact is that everything we learn in math is just a preparatory step for some other topics down the road. The remainder theorem that you are applying to solve the current problem will be needed when you will be solving polynomial equations, factoring polynomials, and solving differential equations, etc. If you do not build a solid foundation now, you will find that math is difficult, later on if not now already. Here's a link to a site that explains the topic in an easy to understand language: http://www.purplemath.com/modules/remaindr.htm