Finding partial fraction expansions

• smalllittle
In summary, the conversation discusses finding the partial fraction expansions of a given polynomial, specifically 10/[(s^2)(s^2+6s+10)]. The attempt at a solution involves expanding the polynomial in the form A/s^2+B/s+(Cs+D)/(s^2+6s+10), with the main idea being that the equation must be true for any value of s except those that make one or more denominators vanish. The conversation concludes with the suggestion to pick four values of s in order to obtain four equations and solve for A, B, C, and D.
smalllittle

Homework Statement

3. Find the partial fraction expansions of the following polynomials

a) 10/[(s^2)(s^2+6s+10)]

The Attempt at a Solution

I assume i should expan the polynomial in the form A/s^2+B/s+(Cs+D)/(s^2+6s+10) but i stuck at finding B. Maybe my expansion is just wrong .. :(

The form of the expansion is correct. It should work.

The main idea is that the equation
$$\frac{A}{s^2} + \frac{B}{s} + \frac{Cs + D}{s^2 + 6s + 10} = \frac{10}{s^2(s^2 + 6s + 10)}$$
has to be true for any value of s except those that make one or more denominators vanish. If you pick four values of s you should get four equations in four unknowns, from which you should be able to get A, B, C, and D.

Have found C?

C and B are related ...

got it. thanks all

1. What is a partial fraction expansion?

A partial fraction expansion is a method used in algebra to simplify a rational function by expressing it as a sum of simpler fractions. It is often used to solve integration problems and to find the inverse of a Laplace transform.

2. When is it necessary to use partial fraction expansion?

Partial fraction expansion is necessary when the degree of the numerator is greater than or equal to the degree of the denominator in a rational function. It is also used when the denominator can be factored into linear and quadratic factors.

3. How do you find the partial fraction expansion of a rational function?

To find the partial fraction expansion, you first need to factor the denominator of the rational function into linear and quadratic factors. Then, you set up a system of equations using the coefficients of the fractions and the original function. Finally, you solve for the unknown coefficients using algebraic methods.

4. Can partial fraction expansion be used for improper fractions?

Yes, partial fraction expansion can also be used for improper fractions. In this case, you first need to perform long division to convert the improper fraction into a polynomial plus a proper fraction. Then, you can use the same method as before to find the partial fraction expansion.

5. What are the applications of partial fraction expansion in science?

Partial fraction expansion has various applications in science, particularly in the fields of physics, engineering, and mathematics. It is used in solving differential equations, finding the inverse Laplace transform, and solving integration problems in calculus. It is also used in signal processing and control systems.

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