Problem 38 - College Algebra

Question

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

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{x^{2}+2x+3}{(x^{2}+4)^{2}}\) is \(\frac{1}{x^{2}+4}+\frac{2x-1}{(x^{2}+4)^{2}}\).

Step 1: Identify the Form of Decomposition

The denominator is \((x^{2}+4)^{2}\) which is a square of a binomial. Consequently, the general form of the decomposition will be \(\frac{A}{x^{2}+4}+\frac{Bx+C}{(x^{2}+4)^{2}}\), where A, B, C are constants to be determined.

Step 2: Equate the Original Expression to General Decomposition Form and Collect Coefficients

We equate the original expression to the form identified in Step 1: \(\frac{x^{2}+2x+3}{(x^{2}+4)^{2}} = \frac{A}{x^{2}+4}+\frac{Bx+C}{(x^{2}+4)^{2}}\). Next, we multiply through by the denominator to get the equation in terms of the coefficients; that would eliminate the fractions. Performing these operations gives us \(x^{2}+2x+3 = A(x^{2}+4) + Bx+C\). Since this equation must hold for all x, the coefficients of the like terms on both sides must be equal.

Step 3: Solve for Coefficients

Matching the coefficients of the similar powers on the two sides of the equation obtained in Step 2 gives a system of equations. Coefficient matching for \(x^{2}\) gives \(1 = A\), for \(x\) gives \(2=B\), and for the constants gives \(3 = 4A+C\). Substituting the coefficient of \(x^{2}\) (i.e., A =1) into the last equation gives \(3 = 4(1) + C\), which implies \(C = -1\). Thus, A = 1, B = 2, C = -1.

Step 4: Write the Partial Fractions Decomposition

Substitute the values of A, B, and C into the general form of the decomposition obtained in Step 1: \(\frac{x^{2}+2x+3}{(x^{2}+4)^{2}} = \frac{1}{x^{2}+4}+\frac{2x-1}{(x^{2}+4)^{2}}\). This is the partial fraction decomposition of the given rational expression.

Key Concepts

Rational Expressions

A rational expression is quite similar to a fraction in arithmetic, except it is composed of polynomials. It has polynomials in both its numerator and its denominator. For instance, \( \frac{x^2+2x+3}{(x^2+4)^2} \) is a rational expression.

• The numerator is \(x^2 + 2x + 3\), which is a polynomial of degree 2.
• The denominator is \((x^2 + 4)^2\), indicating a polynomial raised to a power.

Understanding rational expressions is essential because they allow us to perform algebraic operations like addition, subtraction, multiplication, and division, similarly to how we handle regular fractions. This understanding lays the groundwork for more complex operations like partial fraction decomposition, which expresses a rational expression as a sum of simpler fractions.

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial. It's similar to long division in arithmetic but done with variables.

• The division of polynomials often precedes the simplification of rational expressions.
• Its purpose can be to reduce the complexity of the numerator or denominator of a rational expression, helping with further mathematical operations like factorization.

In the context of partial fraction decomposition, polynomial division helps ensure the numerator degree is less than the denominator degree. If not, division may be necessary to format the expression such that partial fractions can be applied effectively.

Algebraic Fractions

Algebraic fractions look like standard fractions but involve polynomials. They express one polynomial divided by another.

• These fractions can often be simplified by factoring the numerator and the denominator to remove common factors.
• Manipulation of algebraic fractions involves understanding how polynomials interact when combined, subtracted, or decomposed.

Simplifying these fractions is crucial when decomposing them into partial fractions. This process involves expressing the initial complex rational expression, such as the given one \(\frac{x^2+2x+3}{(x^2+4)^2}\), into simpler addend fractions for easier integration or analysis.

Polynomial Equations

Polynomial equations are expressions set to be equal on both sides and contain polynomials. Solving these equations usually involves equating coefficients when both sides contain polynomials.

• This strategy ensures that each term on one side corresponds to a term on the other side.
• From these, you derive several simple algebraic equations where coefficients of like powers must match.

For the exercise given, forming and solving polynomial equations was crucial. By matching coefficients, we find constants A, B, and C, needed for the partial fraction decomposition. These values allow us to express the solution to the rational expression effectively as a sum of simpler rational expressions.