Polynomial Long Division in place of regular division

[KarlyD]

A student decides to divide 4529 by 23 using polynomials, as a substitute for long division. He does this by converting the numbers into polynomials before doing the substitution. He writes numbers as a sum of successive powers of 10; for example, 215=(2 x 10^2) + (1 x 10^1 + 5), and then substitutes x = 10 (in this manner, 215=2x^2 + x + 5)

a) Convert 4529/23 into a problem using division by polynomials, and then perform the division.

b) To complete the division, the student substituted x=10 in the quotient and remainder. Should he get the same quotient and remainder if he had performed simple long division on 4529/23? Why or why not?

c) Does this method have any advantages? Can it be used in other situations?

**I'm confused by this question in its entirety. Why someone would complicate things like that, I don't know. Please help me!! Any assistance would be greatly appreciated.

 

[stapel]

a) They gave you a worded explanation and a worked example of how to do this part. Where are you stuck in the conversion of "4529" and "23" to polynomials? Please be specific!

You say that you are confused by every portion of this exercise. Do you need lessons on polynomial division? :?:

Helping with parts (b) and (c) will have to wait until you're clear on part (a), so let's dig in!

Eliz.

 

[KarlyD]

Absolutely-I have no idea how to go about polynomial division. I've read through my textbooks but still find myself having trouble!

 

[stapel]

Absolutely-I have no idea how to go about polynomial division.

It's discouraging that your class didn't cover this topc before assigning homework which requires it. Fortunately, there are loads of online lessons available, and the topic isn't that complicated. So please pick a few to study:

. . . . .Google results for "polynomial long division"

Once you've studied a few lessons (at least two!), please take your polynomial versions of "4529" and "23", and attempt the division. If you get stuck, or are unsure of your answer, please reply showing your steps.

Thank you!

Eliz.

 

[KarlyD]

I think what threw me off initially was the "1" in the question which is a typo, I believe:
215 = 2 x 10^2 + 1 x 10^1 + 5

Here's what I've done:

23 is
(10^1 + 10^1 +3)

4529 is
(4 x 10^3) + (5 x 10^2) + (10^1 + 10^1 + 9)

so..

(10^1 + 10^1 +3) / (4 x 10^3) + (5 x 10^2) + (10^1 + 10^1 + 9)
(I'm dividing the larger equation by the smaller, though I can't set it up fully via html)

replace 10 with x

(2x + 3) / (4x^3) + (5x^2) + (2x+9)

I'm not sure if I'm going about this properly. To start, you're supposed to look at the 2x and 4x, right? I see that I can divide 2x^2 into it, but I don't know what to do with the 3.

 

[stapel]

I think what threw me off initially was the "1" in the question which is a typo, I believe: 215 = 2 x 10^2 + 1 x 10^1 + 5

Other than dropping the parentheses from (2 × 10²) + (1 × 10¹) + 5, I'm afraid I'm not seeing any difference ("a typo") between what you did and what the book did...? :?:

23 is
(10^1 + 10^1 +3)

...or, in the notation they explained:

. . . . .23 = 20 + 3 = 2(10) + 3 = (2 × 10¹) + 3

4529 is
(4 x 10^3) + (5 x 10^2) + (10^1 + 10^1 + 9)

...or, in the notation they explained:

. . . . .(4 × 10³) + (5 × 10²) + (2 × 10¹) + 9

replace 10 with x

(2x + 3) / (4x^3) + (5x^2) + (2x+9)

Actually, I think you're supposed to be dividing 4529 by 23, not the other way around...? :shock:

. . . . .(4x³ + 5x² + 2x + 9) / (2x + 3)

To start, you're supposed to look at the 2x and 4x, right?

I'm not seeing a "4x" term...?

I believe you're supposed to look at the leading terms. In this case, they would be the 4x³ and the 2x. The lessons (in the link provided earlier) can explain the process in detail.

Eliz.

 

[KarlyD]

I am dividing 4529 by 23 - I explained that I set it up the way I'm writing it on my paper.

That was my mistake - I was referring to 4x^3 instead of 4x for the leading terms. After reading the lessons, I still don't understand what to do with the 3 from 2x + 3.

 

[stapel]

After reading the lessons, I still don't understand what to do with the 3 from 2x + 3.

The online lessons would all have explained and shown the polynomial multiplication. But if you haven't done polynomial division, you probably haven't done the multiplication, either, which is why none of the lessons would have made sense to you. :shock:

Unfortunately, we cannot teach lessons within this format. If all those online lessons didn't do the trick, and your class hasn't covered this yet, then I'm afraid we have have "hit the wall" on what we can accomplish here. Sorry!

You may need to conference with your academic advisor regarding the course requirements, since they seem to be testing you over material that hasn't even begun to be covered. Or else you may need to hire a local tutor. By working face-to-face for a few hours a week (daily would be best), the tutor may be able to teach you the missing course material and get you caught up to the current content. :idea:

Good luck!

Eliz.

 

[KarlyD]

I've already contacted my prof for help, but haven't heard from him yet.

Thanks a lot for your help. :wink:

 

[Deleted member 4993]

so you have:

\(\displaystyle \frac{4x^3 + 5x^2 + 2x + 9}{2x + 3}\\)

first term of your quoteint would be $\displaystyle 2x^2$ since that multiplied by the first term of your divisor (2x) would give you the first term of your dividend ($\displaystyle 4x^3$)

now you have

\(\displaystyle 2x^2\cdot\(2x + 3)\) = $\displaystyle 4x^3 + 6x^2$

subtract this from your given dividend:

$\displaystyle (4x^3 + 5x^2 + 2x + 9) - (4x^3 + 6x^2) = -x^2 + 2x + 9$.............(1)

This is your new dividend.

Next term of your quotient would be -1/2 * x (for reasons explained above)

-1/2 * x * (2x + 3) = - x^2 - 3/2 * x

subtract this from fom your new dividend (1):

(-x^2 + 2 * x + 9) - (- x^2 - 3/2 * x) = 7/2 * x + 9..............(2)

This is your new dividend.

Next term of your quotient would be 7/4 (for reasons explained above)

7/4 * (2x + 3) = 7/2 * x + 21/4

subtract this from fom your new dividend (2):

(7/2 * x + 9) - (7/2 * x + 21/4) = 15/4

So your quotient is 2x^2 - x/2 + 7/4 with a remainder of 15/4

Now check the answer by replacing x with 10