Partial Fractions: Solving Examples with a Novel Method

In summary, the conversation discusses a novel method for expressing fractions in the form of partial fractions, by finding the values of A, B, and C through solving simultaneous equations. This method, known as the Heaviside method, makes integration easier. The conversation also mentions another method, Vedic math, and provides an example of solving for the constants using a system of linear equations.
  • #1
WORLD-HEN
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Hi, me with my really old book again. This time , a novel way of turning expressions into partial fractions.
It would be best if I show you the examples in the book :

[tex] \frac{3x^2 +12x +11} {(x+1)(x+2)(x+3)} [/tex]

To express this fraction in the form

[tex] \frac{A} {x+1} + \frac{B} {x+2} + \frac{C} {x+3} [/tex]

We have to find the values of A , B and C by solving three simultaneous equations ( by the usual method )

The method proposed in the book :

equate the denominator of [tex]A[/tex] to zero and thus get the 'paravartya'
Then substitute this value is the original equation but without the factor which is A's denominator. The result is the value of A.

so for A, the 'paravartya' is -1 ( from x+1 = 0 )
substituting this in
[tex] \frac {3x^2 +12x +11} {(x+2)(x+3)} [/tex]
we get 1, so A = 1

similarly for B, the 'paravartya' is -2 (from x+2 =0)
substituting this in
[tex] \frac{3x^2 +12x +11} {(x+1)(x+3)} [/tex]
we get 1, so B = 1

For C, the paravartya is -3 ( from x+3 =0)
substituting this in
[tex] \frac{3x^2 +12x +11} { (x+1)(x+2) }[/tex]
we get 1, so C = 1

So, the partial fraction is
[tex] \frac{1} {x+1} + \frac{1} {x+2} + \frac{1} {x+3} [/tex]

This makes integration so much easier!
 
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  • #2
That's basically the usual method of 'creating' partial fractions. You should notice that after multiplying the new fractions out, A and B get multiplied by the denominator of C, B and C by that of A, and A and C by that of B. So when you use the "paravartya" of A, the coefficients of B and C become zero, since the "paravartya" sets the denominator of A to zero. Now all that is left is comparing the RHS with the LHS (in particular, the nominators) after using the value of the "paravartya":

[tex]3(-1)^{2}+12(-1)+11 \equiv A(-1+2)(-1+3)[/tex]

[tex]2 \equiv 2A \; \Rightarrow \; A=1[/tex]
 
  • #3
That's Vedic math, isn't it ? :smile:
 
  • #4
yes it is :)
 
  • #5
You can also solve for the constants using a system of linear equations.

[tex] \frac{3x^2 +12x +11} {(x+1)(x+2)(x+3)}=\frac{A}{(x+1)}+\frac{B}{(x+2)} +\frac{C}{(x+3)} [/tex]

Multiply both sides by the common denominator

[tex] 3x^2 +12x +11=A(x+2)(x+3)+B(x+1)(x+3)+C(x+1)(x+2) [/tex]

[tex] 3x^2 +12x +11=A(x^2+5x+6)+B(x^2+4x+3)+C(x^2+3x+2) [/tex]

[tex] 3x^2 +12x +11=Ax^2+5Ax+6A+Bx^2+4Bx+3B+Cx^2+3Cx+2C [/tex]

[tex] 3x^2 +12x +11=(A+B+C)x^2+(5A+4B+3C)x+(6A+3B+2C) [/tex]

The coefficients on either side of the equation must be equal, so

[tex] 3= A+ B+ C [/tex]
[tex] 12=5A+4B+3C [/tex]
[tex] 11=6A+3B+2C [/tex]

Solving this set of equations will get the same results (A=1, B=1, C=1)

Source: http://www.wholikeshomework.com/tutorials/partialfractions.pdf
 
  • #6
It's not Vedic math at all, but known as the Heaviside method, I believe.
 

1. What are partial fractions?

Partial fractions are a method used to simplify complex fractions into smaller, more manageable fractions. They involve breaking down a fraction into smaller fractions with simpler denominators.

2. Why is it important to know how to solve partial fractions?

Solving partial fractions is important in various fields of science, such as engineering and physics, as it allows for the simplification of complicated equations and makes them easier to work with. It also helps in solving integrals and differential equations.

3. Can you provide an example of solving partial fractions using the novel method?

Yes, for example, the fraction 5x / (x^2 + 3x + 2) can be broken down into 2 smaller fractions: 5x / (x + 1) and -5x / (x + 2). This can be solved using the novel method by setting up a system of equations and solving for the unknown coefficients.

4. Are there any special cases to consider when using the novel method for solving partial fractions?

Yes, there are some special cases to consider, such as when the denominator includes repeated factors or when the degree of the numerator is greater than or equal to the degree of the denominator. These cases may require additional steps in the solving process.

5. Is there an alternative method for solving partial fractions?

Yes, there is an alternative method called the Heaviside cover-up method, which involves using a series of algebraic manipulations to solve for the unknown coefficients. However, the novel method is often preferred as it is more efficient and easier to use.

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