# Partial Fraction Decomposition

• jdawg
Sorry about that. In summary, the correct method for solving the given equation is to divide the denominator into the numerator, resulting in the answer t^2(t^2+9). The incorrect method, 1+9/(t^4+9t^2), may be avoided by noting that t^4+9 = t^4+9t^2-9t^2+9.
jdawg

(t4+9)/(t4+9t2)

## The Attempt at a Solution

I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is the answer I got:

1+9/(t4+9t2)

jdawg said:

(t4+9)/(t4+9t2)

## The Attempt at a Solution

I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is the answer I got:

1+9/(t4+9t2)

##t^4+9t^2 = t^2(t^2+9)##.

Thanks! I was worried that I had done it incorrectly, I'm pretty rusty with division with polynomials.

You did do it incorrectly.
$$1+\frac{9}{t^4+9t^2} = \frac{(t^4+9t^2)+9}{t^4+9t^2} \ne \frac{t^4+9}{t^4+9t^2}$$ You can avoid doing long division by noting that ##t^4+9 = t^4+9t^2-9t^2+9##.

## 1. What is Partial Fraction Decomposition?

Partial Fraction Decomposition is a mathematical technique used to break down a rational function (a fraction with polynomials in the numerator and denominator) into smaller, simpler fractions.

## 2. Why is Partial Fraction Decomposition useful?

Partial Fraction Decomposition helps simplify complex algebraic expressions and makes it easier to integrate rational functions. It is also used in solving differential equations and in other areas of mathematics.

## 3. How do you perform Partial Fraction Decomposition?

To perform Partial Fraction Decomposition, the rational function is first factored into irreducible polynomials. Then, each factor in the denominator is expressed as a separate fraction, with a unique polynomial in the numerator. The resulting fractions are added together, and the coefficients of the terms with the same degrees are equated to simplify the expression.

## 4. Can all rational functions be decomposed using Partial Fraction Decomposition?

Yes, all rational functions can be decomposed using Partial Fraction Decomposition, as long as the denominator is factored into distinct irreducible polynomials. If the denominator has repeated factors, additional steps may be required.

## 5. What are some applications of Partial Fraction Decomposition?

Partial Fraction Decomposition is commonly used in calculus, particularly in integration of rational functions. It is also used in solving systems of linear equations, in control theory for analyzing dynamical systems, and in signal processing for representing and manipulating signals.

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