# Long Division and Remainder Theorem

1. Sep 14, 2016

### FritoTaco

NO TEMPLATE BECAUSE MOVED FROM ANOTHER FORUM

Hello,

I've been trying to figure out how it works for complicated problems, I know how to use long division, but I'm not understanding how this process is done for a problem like I have.

Instructions: Write the function in the form ƒ(x) = (x - k)q(x) + r for the given value of k, and demonstrate that ƒ(k) = r

Problem: ƒ(x) = -4x³ + 6x² + 4,
k = 1 - √3 <--- when plugging in q in divisor, signs change as you can see in my picture.​

My Work:
So as you can see from my first attached file, I knew that if I wanted to cancel out 4x³ in the dividend, I would multiply 4x³ on top (quotient) with -1 (divisor). But I also now have to multiply 4x³ with √3. I don't know how to answer it? That's where my question mark is.

In my second attachment, you can see I put what I think is right. Then I drop down the 6x² because that's what you do in long division. So the 4x³ cancels out, but then I have two different degrees in the next part. I don't think I can do anything with that.

Another thing is that you've probably read the "Remainder Theorem in the question, I've already done that here so that's fine. I get a remainder of 0. This is the remainder I should get when I long divide.

Remainder Theorem
ƒ(1 - √3) = -4(1 - √3)³ + 6(1 - √3)² + 12(1 - √3) + 4
Solve: equals 0, so remainder is 0.

Then, for the answer, it's asking to put into form of, ƒ(x) = (x - k)q(x) + r. I already have (x - k) which is the (-1 + √3), I would multiply that with q(x), which is what I'm stuck on for getting the quotient in long division. Lastly, I would add the remainder (r) which there is none.

#### Attached Files:

File size:
33.3 KB
Views:
21
• ###### IMG_0016.JPG
File size:
51.9 KB
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Last edited by a moderator: Sep 14, 2016
2. Sep 14, 2016

### SammyS

Staff Emeritus
A screen shot of your 2nd division attempt:

You need to divide by x - k, which in this case is $\ x-1+\sqrt{3\,}\ .\$ You left out the $\ x\ .$

3. Sep 15, 2016

### FritoTaco

I think I know what you're saying but here is what I have so far.

#### Attached Files:

• ###### IMG_0042.JPG
File size:
42.1 KB
Views:
25
4. Sep 15, 2016

### SammyS

Staff Emeritus

You have unbalanced parentheses and an x2 in the wrong place, or an extra x2.

I suggest you leave the coefficient of x2 intact, that is to say, write the above line as:

$\ -(-4x^3+(4-4\sqrt{3\,})x^2)\$

With long division, keeping track of all those signs gets to be a pain. Do you know synthetic division ?

Last edited: Sep 15, 2016
5. Sep 15, 2016

### FritoTaco

Haha, it's funny how you mention synthetic division just now. I asked my professor today where I left off with you and he said to use synthetic division. Hey, thank you very much for your help, I do appreciate it, it helped me understand more.

#### Attached Files:

• ###### IMG_0045.JPG
File size:
42.5 KB
Views:
111
6. Sep 15, 2016

### SammyS

Staff Emeritus
You are welcome, and thanks for posting your final result. I'll display the final image below: