Polynomial Division: Solving Denominator > Numerator

In summary, the conversation discusses the topic of polynomial division with a larger denominator than numerator. The speaker has been struggling with synthetic division and other methods and asks for help and suggestions. They try using the P/Q method and partial fractions, but make a mistake in their calculations. After correcting their mistake, they are able to solve for A and B and simplify the quotient. However, they encounter another problem when the denominator is a perfect square. After some more attempts, they finally solve for A and B correctly.
  • #1
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I just can't remember how to do this! I've been to several sites suggesting synthetic division and other guides on polynomial division but i can't get it into my head and its driving me wild.

(8x-8)/(x^2+3x+2)

synthetic division doesn't work here because the denominator is larger than the numerator... right?

Heres my work I've done w/ that P/Q method:

(x^2+3x+2) = (x+2)(x+1)

therefore use the format:

8x-8 = A/(x+2) + B(x+1)
and solve for A and B

This is where I am stuck. I went further and solved A = 16 and B = -8 and pluged it into get

16/(x+2) - 8/(x+1) but this doesn't equal (8x-8) / (x^2+3x+2) !

Im stuck! can someone please enlighten me? Thanks a ton!

OR: If anyone has a good helpful website that they know of that shows polynomial division when the Denominator is larger than the numerator id appreciate it. I can only find the sites where the numerator is larger than the denominator
 
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  • #2
Try solving for A and B again! I thin you made a mistake somewhere. :smile:
 
  • #3
What I think you're trying to do is called partial fractions. Polynomial division in this case is very easy: the quotient is 0 and the remainder is 8x-8. :wink:


Your goal, presumably, is to write the quotient

[tex]
\frac{8x - 8}{x^2 + 3x + 2}
[/tex]

in the form

[tex]\frac{A}{x+2} + \frac{B}{x+1}.[/tex]

So you want those to be equal, right? ...

(clearly 8x-8 = A/(x+2) + B(x+1) was a typo -- what did you really mean by that?)
 
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  • #4
my goal was to simplify it enough so i can integrate it easially. I am going to work on it some more right now and ill report back.
 
  • #5
Ok here's my new work:

8x-8 = A(x+2) + B(x+1)

Sub in -2 for x to get B
8(-2)-8 = A(-2+2) + B(-2+1)

Simplify to
B = 24

then i subed in -1 for x to solve for A
8(-1)-8 = A(-1+2) + B(-1+1)

simplify to
A = 16 <------------ Thats where my mistake was! its -16 not 16!

OMG YAY YAY YAY YAY YAY I CAN DO MATH!

THANKYOU ALL SO MUCH

But this brings up another question. I tried to do a problem with a perfect square in the denominator and the method doesn't work... :-(

This is the one i made right now

(8x+12) / (x+1)^2

so:
[(x+1) (x+1)^2 (8x+12)] / (x+1)^2 = [A/(x+1) + B/ (x+1)^2 (x+1)(x+1)^2]

simplifies to

(x+1) (8x+12) = A(x+1)^2 +B(x+1)

when i try to solve for A and B, no matter what number i sub in for X they both end up zeroing out and I am left with nothing!

Any insite this time? Thanks so much so far
 
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  • #6
Ok in my attempt i multiplied both sides by both terms. I tried it by multiplying both sides by one term at a time and i think its working... ill report back
 
  • #7
Nope doesn't work. I am stuck again :-( I ended getting A = 0 and B = 4. A = 0 doesn't really make a whole lotta sense.
 
  • #8
Actually I just figured it out. I learned that my alegebra needs some work.

A = 8
B = 4
THANKS ill be back later with more questions!
 

1. What is polynomial division?

Polynomial division is a method of dividing two polynomials, where the denominator (the polynomial being divided by) has a higher degree than the numerator (the polynomial being divided). It involves using long division or synthetic division to find the quotient and remainder.

2. How do you solve for the quotient and remainder in polynomial division?

To solve for the quotient and remainder in polynomial division, you can use either long division or synthetic division. In long division, you divide the terms of the numerator by the first term of the denominator, then multiply the result by the entire denominator and subtract from the numerator. The resulting number is the remainder. In synthetic division, you use a shortcut method that involves only the coefficients of the polynomials and does not require writing out all the terms.

3. What are some common mistakes to avoid when solving polynomial division?

Some common mistakes to avoid when solving polynomial division include forgetting to change the signs when subtracting, not lining up the terms correctly, and forgetting to bring down the next term in the numerator after each step of the division. It is also important to double check your final answer by multiplying the quotient by the denominator and adding the remainder to ensure it equals the numerator.

4. What is the degree of the resulting quotient in polynomial division?

The degree of the resulting quotient in polynomial division is equal to the difference in degree between the numerator and denominator. For example, if the numerator has a degree of 3 and the denominator has a degree of 2, the resulting quotient will have a degree of 1.

5. Can polynomial division be used to solve equations?

Yes, polynomial division can be used to solve equations that involve a denominator that is a polynomial. This is because dividing by a polynomial is equivalent to multiplying by its reciprocal. By dividing both sides of the equation by the polynomial, you can isolate the variable on one side and solve for it.

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