Problem 11 - Precalculus Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x^{3}+x+1}{x(2 x-5)^{3}\left(x^{2}+2 x+5\right)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is: \( \frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex+F}{x^2+2x+5} + \frac{Gx+H}{(x^2+2x+5)^2} \).

Step 1: Identify Denominator Factors

First, recognize the distinct factors in the denominator: \(x\), \((2x-5)^3\), and \((x^2+2x+5)^2\). These will form the basis for the partial fraction decomposition.

Step 2: Assign Simple Linear Factor

For the linear factor \(x\), assign a term \(\frac{A}{x}\), where \(A\) is a constant to be determined.

Step 3: Assign Repeated Linear Factor

For the repeated linear factor \((2x-5)^3\), assign terms for each power up to the third: \(\frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3}\), where \(B\), \(C\), and \(D\) are constants to be determined.

Step 4: Assign Quadratic Factor

For the quadratic factor \((x^2+2x+5)^2\), assign terms for each power up to the second. Since the factor is a quadratic, each term will have a linear numerator: \(\frac{Ex + F}{x^2+2x+5} + \frac{Gx+H}{(x^2+2x+5)^2}\), where \(E\), \(F\), \(G\), and \(H\) are constants to be determined.

Step 5: Combine Terms for General Form

Combine all the terms from Steps 2, 3, and 4. The general form of the partial fraction decomposition is: \[ \frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex + F}{x^2+2x+5} + \frac{Gx + H}{(x^2+2x+5)^2} \]

Key Concepts

Polynomial Division

Before diving into partial fraction decomposition, it's useful to understand polynomial division—a crucial technique when dealing with rational expressions. Polynomial division is similar to long division with numbers, but it involves dividing polynomials. When you have a polynomial divided by another, you can use either long division or synthetic division to simplify the expression.

Long division is like the regular division method but involves handling the polynomial terms. It helps to simplify expressions and reveal the remainder. The key steps are:

• Divide the highest degree term of the dividend by the highest degree term of the divisor.
• Multiply the entire divisor by this result and subtract from the dividend.
• Repeat the process with the remainder.

Polynomial division helps you determine whether a rational expression can be simplified and sometimes helps in setting up partial fraction decompositions.

Linear Factors

Linear factors are expressions of the form \(ax + b\). They are simple to solve and play an essential role in partial fraction decomposition because each linear factor in the denominator is assigned a term with a constant numerator.

In partial fraction decomposition:

• For each linear factor in the denominator, like \(x\), allocate a fraction with an unknown constant numerator, such as \(\frac{A}{x}\).
• If you have repeated linear factors, extend the decomposition by assigning terms for each power, up to the highest exponent.

By breaking down these factors, solving for the unknown constants becomes easier as they form simple algebraic equations.

Quadratic Factors

Quadratic factors are expressions like \(ax^2 + bx + c\). These can be more complex than linear factors, particularly if they do not factor easily into linear expressions.

In partial fraction decomposition, each quadratic factor entails a linear numerator:

• For a quadratic factor, allocate fractions with numerators in linear form, such as \(Ex + F\).
• Handle repeated quadratic factors by assigning terms for each power of the factor, up to its highest exponent.

Understanding quadratic factors is crucial in ensuring that each factor in the original effect is represented properly in the decomposition, assisting in solving for unknowns comprehensively.

Repeated Roots

Repeated roots occur when a factor appears more than once in the polynomial expression's factorization. In partial fraction decomposition, they need particular attention.

To handle repeated roots:

• For a root repeated \(n\) times, such as \((2x-5)^3\), allocate separate terms for each degree from 1 to \(n\): \(\frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3}\).
• This ensures each power of the repeated factor is accounted for, allowing you to unravel the complexity of the rational expression.

Repeated roots expand the scope of decomposition and often need careful computation to determine the specific constants involved.