Problem 29 - Precalculus Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{2 x}{4 x^{2}+12 x+9}$$

Step-by-Step Solution

Answer

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

Step 1: Factor the Denominator

First, identify the form of the denominator to check if it can be factored. The denominator of the given rational function is \(4x^2 + 12x + 9\). This is a quadratic expression and can potentially be factored into two binomials. After factoring, we get \((2x + 3)(2x + 3)\) or \((2x + 3)^2\).

Step 2: Set up the Partial Fractions

Since the denominator factors to \((2x + 3)^2\), the partial fraction form will involve terms with \((2x + 3)\) and \((2x + 3)^2\). Thus, set up the partial fractions as: \[ \frac{2x}{4x^2 + 12x + 9} = \frac{A}{2x + 3} + \frac{B}{(2x + 3)^2} \]

Step 3: Clear the Denominator

Multiply both sides by \((2x + 3)^2\) to eliminate the denominator on the left side of the equation:\[ 2x = A(2x + 3) + B \]

Step 4: Solve for Coefficients A and B

Expand the equation: \[ 2x = 2Ax + 3A + B \].Now, equate the coefficients from both sides of the equation. This gives the system of equations: 1. \(2A = 2\) 2. \(3A + B = 0\)Solve for \(A\) and \(B\). From the first equation, \(A = 1\). Substitute \(A\) into the second equation to find \(B\):\[ 3(1) + B = 0 \Rightarrow B = -3 \].

Step 5: Write the Partial Fraction Decomposition

Now that we have\(A = 1\) and \(B = -3\), substitute back into the partial fractions setup:\[ \frac{2x}{4x^2 + 12x + 9} = \frac{1}{2x + 3} - \frac{3}{(2x + 3)^2} \].

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2. They generally take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Quadratic expressions can represent a wide range of mathematical models, including projectile motion, and are essential in solving many algebraic problems.

In our specific example, we have the quadratic expression \(4x^2 + 12x + 9\). This expression is part of the denominator of the rational function we're decomposing into partial fractions. Recognizing quadratic expressions is vital because they dictate how we approach factorization, one of the key steps in partial fraction decomposition.

Being familiar with the standard forms of quadratic expressions also aids in predicting what type of roots the expression might have, which influences factorization.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into a product of its simpler components, known as factors. This process is crucial, especially when dealing with quadratic expressions, as it can simplify expressions and help in solving equations.

When factoring quadratic polynomials, you look for two binomials that, when multiplied together, give the original quadratic polynomial. In the given exercise, the quadratic expression \(4x^2 + 12x + 9\) factors into \((2x + 3)(2x + 3)\) or \((2x + 3)^2\). Recognizing this perfect square trinomial significantly simplifies the task of setting up partial fractions.

Here are some tips for factoring:

• Look for common factors in all terms and factor them out if possible.
• Use techniques such as grouping, finding a perfect square, or employing the quadratic formula if the expression does not factor neatly.
• Always check by expanding to ensure the factors multiply to the original polynomial.

Understanding these aspects of factoring is essential for effectively tackling polynomial equations.

System of Equations

A system of equations is a set of two or more equations involving the same set of variables. In partial fraction decomposition, systems of equations emerge when equating coefficients to solve for unknown constants, such as \(A\) and \(B\) in our exercise.

In step 4 of the solution, we set up a system of equations: 1. \(2A = 2\) 2. \(3A + B = 0\).

Solving this system involves finding the values of \(A\) and \(B\) that satisfy both conditions. For this problem, solving the first equation is straightforward, yielding \(A = 1\). Substituting this value into the second equation gives \(B = -3\).

Tips for solving systems of equations:

• Use substitution when one equation is easily solvable for one variable.
• Consider elimination when coefficients allow variables to be canceled out by addition or subtraction.
• Check your solutions by substituting them back into the original equations.

Mastering systems of equations is key to finding constant terms in partial fraction decomposition and solving simultaneous algebraic equations efficiently.