Question

Find the partial fraction decomposition of each rational expression. $$ \frac{x^{2}}{\left(x^{2}+4\right)^{3}} $$

Short answer

\( \frac{x^2}{(x^2+4)^3} = \frac{1}{(x^2 + 4)^2} - \frac{4}{(x^2 + 4)^3} \)

Step-by-step solution

- Identify the Form

Recognize that the rational expression is of the form \( \frac{P(x)}{Q(x)} \) where \( P(x) = x^2 \) and \( Q(x) = (x^2 + 4)^3 \).

- Set up the Partial Fraction Decomposition

Express \( \frac{x^2}{(x^2 + 4)^3} \) in terms of partial fractions by assuming the form: \[ \frac{x^2}{(x^2 + 4)^3} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2} + \frac{Ex + F}{(x^2 + 4)^3} \]

- Multiply by the Denominator

Multiply both sides of the equation by \( (x^2 + 4)^3 \) to get: \[ x^2 = (Ax + B)(x^2 + 4)^2 + (Cx + D)(x^2 + 4) + (Ex + F) \]

- Expand and Combine Terms

Expand the right-hand side completely and combine like terms. Then equate the coefficients of corresponding powers of \( x \).

- Solve for Constants

By matching the coefficients from both sides, solve for the constants. Since the left-hand side is \( x^2 \), compare each power of \( x \) and set the coefficients equal to zero to solve for \( A, B, C, D, E, \) and \( F \).

- Form the Decomposed Expression

After finding the constants, substitute them back into the partial fraction form. For this particular example, the solution will be: \[ \frac{x^2}{(x^2 + 4)^3} = \frac{1}{(x^2 + 4)^2} - \frac{4}{(x^2 + 4)^3} \]

Key concepts

rational expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Rational expressions are often given in a form that can be broken down into simpler fractions, which makes calculations easier. For example, consider \(\frac{x^{2}}{(x^2 + 4)^3}\), where the numerator is a polynomial \(x^2\), and the denominator is a polynomial \(\(x^2 + 4\)^3\). Decomposing the rational expression helps in understanding and solving more complex mathematical problems.

polynomial long division

Sometimes, before we can decompose a rational expression into partial fractions, we need to simplify it using polynomial long division. However, in this case, the numerator's degree (\(2\)) is lower than the denominator's degree (\((x^2 + 4)^3\)) multiplied, so polynomial long division isn't necessary. Polynomial long division is a process similar to numerical long division, applied to polynomials. It's used when the degree of the numerator is higher than or equal to the degree of the denominator.

terms expansion

To find the partial fractions, we multiply out both sides by the common denominator to clear the fraction. For our example \(\frac{x^2}{(x^2 + 4)^3}\), we assumed the form \(\frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2} + \frac{Ex + F}{(x^2 + 4)^3}\). We then multiply both sides by \((x^2 + 4)^3\) to get rid of the denominators:

• Multiply: \(x^2 = (Ax + B)(x^2 + 4)^2 + (Cx + D)(x^2 + 4) + (Ex + F)\)
• Expand each term: \( (Ax + B)(x^2 + 4)^2 = (Ax + B)(x^4 + 8x^2 + 16)\)
• Combine like terms: Match & combine the coefficients of each power of \x\ to simplify the equation.

finding constants

After expanding and combining terms, we need to determine the constants A, B, C, D, E, and F. Given our expanded equation: \(x^2 = (Ax + B)(x^4 + 8x^2 +16) + (Cx + D)(x^2 + 4) + (Ex + F)\), we compare the coefficients of corresponding powers of \x\ on both sides. This results in a system of equations:

• For the \x^4\ term: Coefficient of \x^4\ on the left side = Coefficient of \x^4\ on the right side (A's influence).
• For the \x^3\ term: Coefficient of \x^3\ on the left side = Coefficient of \x^3\ on the right side (No x^3 term in our case).
• For lower terms: Continue comparing all power terms (x, constant terms).

Solve this system to find A, B, C, D, E, and F. In our example, we find A, B, C, D, E, and F such that the final decomposed expression is:
\( \frac{x^2}{(x^2 + 4)^3} = \frac{1}{(x^2 + 4)^2} - \frac{4}{(x^2 + 4)^3} \)