Problem 28 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

In solving such a rational function decomposition, it's crucial to identify the form the decomposition should take and then equate it to the original rational expression. This will produce polynomial equations through which we can solve for the unknown coefficients. The coefficients then substitute into the form identified in the first step, giving the decomposition.

Step 1: Recognize the form of the Rational Expression

The given expression can be written as \( \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{C}{(x+1)} + \frac{D}{(x+1)^2} \) where A, B, C, and D are coefficients to be determined.

Step 2: Equate to the Original Rational Expression

Equating this to the original rational expression gives \(x^{2} = A(x-1)(x+1)^2 + B(x+1)^2 + C(x-1)^2(x+1) + D(x-1)^2\). This results in a polynomial equation where the coefficients of like terms on the right hand side must equal those on the left hand side.

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

In order to make the process of solving for A, B, C and D easier, we can choose smart values for x and then solve the equations. That way we can make some terms disappear. For example if we choose x=1, the equation simplifies to \(1^2 = B(1+1)^2\), which we can solve to get B. We can choose values for x that simplify our expression in a similar way to find A, C and D.

Step 4: Write Down the Partial Fraction Decomposition

After finding the values of coefficients A, B, C, and D, substitute these into the initial form obtained in step 1. This will form the partial fraction decomposition of the rational function.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Understanding these expressions is key in algebra, especially when performing algebraic operations such as addition, subtraction, multiplication, or division. When you're dealing with complex fractions containing polynomials with higher degrees, it's often useful to break them down into simpler parts. This process is called partial fraction decomposition.
By decomposing a complex rational expression, we can rewrite it as a sum of simpler fractions, which makes further algebraic manipulation much easier. For example, when integrating rational functions in calculus, partial fractions allow for simpler antiderivatives. Thus, mastering rational expressions and their decomposition is an invaluable skill in algebra.

Coefficients

Coefficients are the numbers or constants that multiply the variables or powers of variables in a polynomial expression. In the context of partial fraction decomposition, coefficients play a vital role in expressing the rational expression as a sum of simpler fractions.
When decomposing a rational expression, you express it in terms of coefficients such as A, B, C, and D in a fractional form. These coefficients are unknowns that represent parts of the whole expression. To find their values, you set up an equation by expanding the sum of partial fractions and equating it to the original expression.
Solving these equations involves comparing the coefficients of like terms on both sides. By choosing strategic values of the variable, you can solve for these coefficients step-by-step, ensuring your partial fraction decomposition accurately represents the original rational expression.

Polynomial Equations

Polynomial equations involve expressions that set a polynomial equal to another, often zero. In partial fraction decomposition, forming polynomial equations is necessary to identify the coefficients in the expression.
After setting up your expression in terms of partial fractions, expand and simplify it until you have a polynomial on both sides of the equation. The degree of the polynomial should match on both sides, as they were initially equal. Each coefficient from the original polynomial must be equal to the sum of the corresponding coefficients from the decomposed expression.
Once the polynomial equation is set up, the task is to solve for the coefficients. It involves careful analysis and selection of specific values for the variable to simplify the equation. This method makes it more straightforward to solve for the unknowns and rewrite the expression as a sum of partial fractions.

Algebraic Simplification

Algebraic simplification is the process of finding a simpler form of an algebraic expression while keeping its value the same. For partial fraction decomposition, simplification helps convert a complex rational expression into a more manageable form.
By separating the original rational expression into partial fractions, you effectively simplify the expression without altering its original value. This makes it easier to work with for further mathematical computations, such as integration or solving equations.
Simplification typically involves breaking down the expression, as seen in our decomposition process. Here, you reduce a single complex fraction into multiple simpler fractions. These stages involve recognizing and manipulating algebraic structures, which aids in promoting clearer understanding and easier problem-solving. Overall, algebraic simplification is key in making complex algebra more approachable.