Problem 34 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)}$$

Step-by-Step Solution

Answer

The final partial fractions decomposition would look like \( \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{Cx + D}{x^{2} + 2}=\frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)}\), with A, B, C and D values computed in Step 3.

Step 1: Identify the correct form of the partial fractions

Looking at the denominator of the given rational expression \( \frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)} \), we can infer the form of partial fractions. The denominator can be rewritten as \( (x-1)^{2} \) and \(x^{2}+2\). Hence, we will have the partial fractions in the form of \( \frac{A}{(x-1)} \), \( \frac{B}{(x-1)^2} \) and \( \frac{Cx + D}{x^{2} + 2}\).

Step 2: Setup the equation

The given rational function can be written as: \( \frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)} = \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{Cx + D}{x^{2} + 2}\. Multiply each side by the denominator \( (x-1)^{2}\left(x^{2}+2\right)\) to get rid of the fraction.

Step 3: Solve for A, B, C and D

Solving for A, B, C and D will involve the comparison of the coefficients on each side of equation from step 2. We may also need to select smart values for x that helps to simplify the calculation and isolate each unknown. This step mainly involves algebraic manipulation, comparison of coefficients and solving the system of linear equations we get.

Step 4: Conclude the final partial fractions decomposition

Once we solve for A, B, C, and D, we substitute these values in the partial fractions form from Step 1. That would give us the final partial fractions decomposition

Key Concepts

Rational Expressions

A rational expression is an algebraic fraction whose numerator and denominator are polynomials. The expression you're trying to decompose, \( \frac{10x^2+2x}{(x-1)^2(x^2+2)} \), is a complex rational expression because its denominator has more than one term.

Understanding partial fraction decomposition is essential for working with such expressions. This process involves breaking down a complex rational expression into simpler fractions that are easier to integrate or differentiate if you're dealing with calculus, or to work with in algebra.

Key steps include identifying unique factors in the denominator and assigning constants to them, in this case, the \( A \) and \( B \) for the \( (x-1) \) factors and \( Cx+D \) for the \( x^2+2 \) factor. The goal is to rewrite the original expression as a sum of these simpler fractions, making other operations with the expression more manageable.

Algebraic Manipulation

Algebraic manipulation is an art of rewriting expressions in a form that makes them easier to understand or solve. In our partial fraction decomposition, this skill is central to finding the values of \( A \) , \( B \) , \( C \) , and \( D \) that make the decomposition work.

Starting with the multiplication of both sides of the equation by the original denominator, you eliminate the denominators entirely. This leaves you with a polynomial that you can equate to the original numerator. The challenge now is to find the values of our unknown constants that will make the equation true for all values of \( x \).

You achieve this by equating coefficients of like powers of \( x \) on both sides of the equation or by substituting strategic values of \( x \) to create simpler equations. Each step requires careful manipulation, such as expanding the expressions, combining like terms, and comparing coefficients across both sides of the equation.

Linear Equations

The finale of partial fraction decomposition often involves solving linear equations. After the algebraic juggling mentioned earlier, you'll end up with a system of equations for \( A \), \( B \), \( C \) and \( D \).

Each equation is linear, meaning that the variables (in our case, the constants) do not have any exponents other than 1. Solving this system of equations requires understanding how to isolate variables, combine like terms, and use substitution or elimination methods to find the constants' values.

Such skills are fundamental not just in algebra, but also in higher-level math and sciences, as linear equations model many real-world phenomena. The ability to navigate these equations will aid in uncovering the values needed to complete the partial fraction decomposition, thus simplifying our complex rational expression.