Problem 28 - Algebra and Trigonometry

Question

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{3 x^{2}+5 x-13}{(3 x+2)\left(x^{2}-4 x+4\right)} $$

Step-by-Step Solution

Answer

The partial fractions are \( \frac{A}{3x+2} + \frac{B}{x-2} + \frac{C}{(x-2)^2} \).

Step 1: Identify the Denominator Components

First, we identify the components of the denominator. The denominator is \((3x+2)(x^2-4x+4)\). Notice that \(x^2-4x+4\) can be factored further as \((x-2)^2\). So the denominator becomes \((3x+2)(x-2)^2\).

Step 2: Set Up the Partial Fraction Decomposition

The rational function can be decomposed into partial fractions. We write:\[\frac{3x^2 + 5x - 13}{(3x+2)(x-2)^2} = \frac{A}{3x+2} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\]where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Combine the Fractions

To find \(A\), \(B\), and \(C\), we re-combine the fractions over a common denominator:\[\frac{A(x-2)^2 + B(3x+2)(x-2) + C(3x+2)}{(3x+2)(x-2)^2}\]This must equal \(\frac{3x^2 + 5x - 13}{(3x+2)(x-2)^2}\). The numerators must be equal for all \(x\).

Step 4: Expand and Rearrange Terms

Expand the numerator:\[A(x^2 - 4x + 4) + B(3x^2 - 4x - 2) + C(3x + 2)\]Combine like terms to form:\[(A + 3B)x^2 + (-4A - 4B + 3C)x + (4A - 2B + 2C)\]

Step 5: Set Up System of Equations

Equate the coefficients from the expanded form to those in the numerator \(3x^2 + 5x - 13\):1. \(A + 3B = 3\)2. \(-4A - 4B + 3C = 5\)3. \(4A - 2B + 2C = -13\)

Step 6: Solve the System of Equations

We solve the system:1. From equation 1: \(A = 3 - 3B\)2. Substitute \(A = 3 - 3B\) into equations 2 and 3 and solve for \(B\) and \(C\) using substitution or elimination. - After solving, find specific values for \(A\), \(B\), and \(C\).

Step 7: Write the Final Decomposition

Substitute the solved values of \(A\), \(B\), and \(C\) into the partial fraction decomposition:\[\frac{A}{3x+2} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\]This gives the final form of the partial fractions.

Key Concepts

Rational Functions

A rational function is essentially a fraction, but instead of simple numbers in the numerator and denominator, it has polynomials. You can think of it like a mathematical sandwich, with polynomials as the bread and the fraction as the filling.

Rational functions are written as \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero. These functions are key players in calculus and algebra.

• The numerator \(P(x)\) could be any polynomial, for example \(3x^2 + 5x - 13\).
• The denominator \(Q(x)\) is another polynomial, such as \((3x+2)(x^2-4x+4)\) from our example.

Rational functions can look complex, but they follow the straightforward rule of fractions: never divide by zero. That makes factoring in the denominator very important.

Factoring Polynomials

Factoring polynomials is a critical skill when working with rational functions, especially when performing partial fraction decomposition.

The denominator in our exercise can be factored into two parts: \((3x+2)\) and \((x-2)^2\).

• First, look for common factors in the polynomial.
• For quadratics like \(x^2 - 4x + 4\), see if they can be simplified further. In this case, it simplifies to \((x-2)^2\).

Properly factoring the polynomial helps us set up the partial fraction decomposition. By breaking down complex polynomials into simpler factors, you ease the path to solving rational function problems. Using these factors, we can express the rational function as a sum of simpler fractions.

System of Equations

When decomposing rational functions, you often end up working with a system of equations. This might sound complex, but it's simply a way to find the values of unknowns in an interconnected setup.

In our example, we determine constants \(A\), \(B\), and \(C\) by setting up equations based on the numerators of matching fractions:

• Equation 1: \(A + 3B = 3\)
• Equation 2: \(-4A - 4B + 3C = 5\)
• Equation 3: \(4A - 2B + 2C = -13\)

Solving these equations requires substituting expressions and employing either substitution or elimination methods. You solve them step by step to find the exact values needed for the final decomposition.

Mastering the system of equations is essential for partial fraction decomposition. This method offers a systematic way to approach and solve algebra problems that involve rational functions.

Numerator and Denominator

Understanding numerators and denominators is fundamental in dealing with rational functions. The numerator is the top part of the fraction, whereas the denominator is the bottom part. These parts determine the fraction's value and are crucial when performing operations such as partial fraction decomposition.

In the exercise, our rational function has a numerator of \(3x^2 + 5x - 13\) and a denominator of \((3x+2)(x-2)^2\).

• The numerator can affect the shape and intercepts of the graph of the function.
• The denominator establishes restrictions, indicating values of \(x\) that make the function undefined.

By matching coefficients from numerators of equal fractions in partial fraction decomposition, we ensure consistency across the equation. Recognizing the role of each component aids in simplifying and reassembling rational functions effectively.