Solving Partial Fractions with Polynomial Division

In summary, the conversation discusses the use of long division in solving the given integral equation. It is mentioned that long division was used to simplify the equation and a link to a video tutorial is provided. An alternative method using substitution is also suggested.
  • #1
beaf123
41
0

Homework Statement



∫ (x^3)/(x^2+2x+1)

I think I could solve it if I knew how they did this operation:

From the solution:
'
(x^3)/(x^2+2x+1) = (x-2) + (3x+2)/(x+1)^2 ( After long division)

Did they use polynomialdivision?

x^3: x^2-2X+1=

If so, how?
 
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  • #2
Yes, long division was used. Do you know how to use it?

This video will help http://www.youtube.com/watch?v=ok4k6HDCuIk and just keep in mind that when applying it to your problem,

[tex]x^3=1x^3+0x^2+0x+0[/tex]
 
  • #3
beaf123 said:

Homework Statement



∫ (x^3)/(x^2+2x+1) dx

I think I could solve it if I knew how they did this operation:

From the solution:
'
(x^3)/(x^2+2x+1) = (x-2) + (3x+2)/(x+1)^2 ( After long division)

Did they use polynomial division?

x^3: x^2-2X+1=

If so, how?

Yes, they used long division for polynomials.

Here's a link to Wikipedia's page on polynomial long division.

If you don't want to use long division, use the substitution u = x+1.
 

1. What is polynomial division?

Polynomial division is a method used to divide polynomials, which are algebraic expressions with one or more terms that can include variables, constants, and exponents. It involves breaking down a polynomial into simpler, smaller parts.

2. How do you solve partial fractions with polynomial division?

To solve partial fractions with polynomial division, you first need to factor the denominator of the fraction into its irreducible factors. Then, you set up a system of equations with the partial fractions on one side and the original fraction on the other side. Finally, you solve for the unknown coefficients by equating the corresponding terms on both sides of the equation.

3. Why do we use polynomial division in partial fractions?

Polynomial division is used in partial fractions because it allows us to break down a complex fraction into smaller, simpler fractions. This makes it easier to integrate or manipulate the original fraction in algebraic equations.

4. What are the steps involved in solving partial fractions with polynomial division?

The steps involved in solving partial fractions with polynomial division are: 1) Factor the denominator of the fraction into its irreducible factors. 2) Set up a system of equations with the partial fractions on one side and the original fraction on the other side. 3) Solve for the unknown coefficients by equating the corresponding terms on both sides of the equation.

5. Can you use polynomial division to solve any type of fraction?

No, polynomial division can only be used to solve fractions where the denominator is a polynomial. It is not applicable to fractions with other types of expressions, such as radicals or trigonometric functions.

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