Problem 31 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{5 x^{2}+6 x+3}{(x+1)\left(x^{2}+2 x+2\right)} $$

Step-by-Step Solution

Answer

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

Step 1: Write the General Form

The given function has a linear term \(x + 1\) and a quadratic term \(x^2 + 2x + 2\). The general form will correspond to this separation. The general form is therefore \(\frac{A}{x+1} + \frac{Bx+C}{x^2+2x+2}\).

Step 2: Clear the Fraction

To simplify the equation, clear the fraction by multiplying the entire equation by the denominator from the left side. This results in: \(5x^2 + 6x + 3 = A(x^2 + 2x + 2) + (Bx + C)(x + 1)\).

Step 3: Expand and Collect Like Terms

In this step, expand the right hand side of the equation and then collect like terms to isolate terms with the same powers of \(x\). This results in: \(5x^2 + 6x + 3= (A + B)x^2 + (2A + B + C)x + (2A + C)\).

Step 4: Equate Coefficients and Solve system of equations

By equating the coefficients of the power terms on both sides of the equation, one can create a system of equations for A, B, and C. The corresponding equations will be \(A + B = 5\), \(2A + B + C = 6\), and \(2A + C = 3\). Solving this system of equations, one finds \(A = 1\), \(B = 4\), and \(C = -1\).

Step 5: Write the Final Answer

Substitute the values for A, B, and C in the general form to get the decomposition. Therefore, the decomposition of the given function is \(\frac{1}{x+1} + \frac{4x - 1}{x^2 + 2x + 2}\).

Key Concepts

Linear Term

In algebra, a linear term refers to an expression that represents a straight line when graphed on a coordinate plane. More specifically, it’s any term in an algebraic expression that contains the variable to the first power, typically written in the form of ax + b, where a is the coefficient and b is the constant. In the context of partial fraction decomposition, a linear term such as x + 1 in the denominator indicates the presence of a unique fraction in the decomposition where the numerator is a constant, represented as \(\frac{A}{x+1}\).
To reveal the value of the constant A, it's necessary to perform algebraic manipulations which often include equating coefficients and solving a system of equations. The value of A should simplify the expression and help it match the original rational expression when the partial fractions are combined.

Importance in Partial Fraction Decomposition

In partial fraction decomposition, the linear term is responsible for breaking down complex fractions into simpler ones which are easier to work with, especially in integration. This fundamental aspect of the technique allows for smoother and more manageable calculation steps.

Quadratic Term

A quadratic term in algebra is a second-degree polynomial which can be written in the form ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. This term is the hallmark of quadratic functions, which graph as a parabola.
In our partial fraction decomposition exercise, the quadratic term is x^2 + 2x + 2. When it appears in the denominator of a rational expression, it signifies that the corresponding fraction in the decomposition will have a linear numerator, usually expressed as \(\frac{Bx + C}{x^2 + 2x + 2}\). Unlike with linear terms, the numerators here involve not just a single constant but a linear expression, because this matches the degree of the denominator minus one.

Role in Decomposition Process

The quadratic term guides us in determining the appropriate form for the partial fractions. It ensures that the decomposition will account for all possible variations in the numerator that can occur when decomposing an expression with a quadratic term. Finding the correct values for B and C is essential to match the original polynomial once the fractions are added together.

System of Equations

A system of equations is a set of multiple equations that share common variables. The solution to a system of equations is the set of values for the variables that satisfies all equations simultaneously. In algebra, solving a system of equations is a method to find the point(s) at which the equations intersect.
During the step-by-step solution of the partial fraction decomposition, after equating coefficients and expanding, we form a system of equations to solve for A, B, and C which are the constants in the numerators of the partial fractions. For our specific exercise, the equations are \(A + B = 5\), \(2A + B + C = 6\), and \(2A + C = 3\), which must be solved together because their solutions are interdependent. By solving this system of equations rigorously, we find the values that satisfy all equations.

Ease of Solution

Using techniques such as substitution, elimination, or matrix operations, we can find the values of the variables that will provide the complete decomposition of the original expression into partial fractions. The process enhances our understanding of the relationship between algebraic expressions and their geometrical representations. It's also an excellent example of how algebraic manipulation and critical thinking come together to solve complex mathematical problems.