Problem 45 - Algebra and Trigonometry

Question

Find the partial fraction decomposition: \(\frac{3 x^{2}+15 x+5}{x^{3}+2 x^{2}+x}\)

Step-by-Step Solution

Answer

\( \frac{3 x^{2}+15 x+5}{x (x + 1)^2} = \frac{5}{x} - \frac{2}{x + 1} + \frac{7}{(x + 1)^2} \)

Step 1: Factor the Denominator

First, factor the denominator of the given fraction. The denominator is: \[ x^3 + 2x^2 + x \]We can factor out an \( x \) from every term:\[ x (x^2 + 2x + 1) \]Notice that \( x^2 + 2x + 1 \) is a perfect square trinomial and can be factored further:\[ x (x + 1)(x + 1) = x (x + 1)^2 \]So, the denominator factors to \( x (x + 1)^2 \).

Step 2: Set Up Partial Fractions

Write the partial fraction decomposition based on the factored denominator. Since the denominator is \( x (x + 1)^2 \), set up the decomposition as:\[ \frac{3 x^{2}+15 x+5}{x (x + 1)^2} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2} \]

Step 3: Combine the Right-Hand Side

Combine the fractions on the right-hand side to have a common denominator:\[ \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2} = \frac{A (x + 1)^2 + B x (x + 1) + C x}{x (x + 1)^2} \]

Step 4: Equate the Numerators

Equate the numerators of the left and right sides:\[ 3x^2 + 15x + 5 = A(x + 1)^2 + Bx(x + 1) + Cx \]Expand the right-hand side:\[ 3x^2 + 15x + 5 = A(x^2 + 2x + 1) + Bx^2 + Bx + Cx \]\[ 3x^2 + 15x + 5 = Ax^2 + 2Ax + A + Bx^2 + Bx + Cx \]Combine like terms:\[ 3x^2 + 15x + 5 = (A + B)x^2 + (2A + B + C)x + A \]

Step 5: Solve for Constants

Compare coefficients for each power of \( x \) to find \( A \), \( B \), and \( C \):From \( x^2 \) term:\[ A + B = 3 \]From \( x \) term:\[ 2A + B + C = 15 \]From constant term:\[ A = 5 \]Substitute \( A = 5 \) into \( A + B = 3 \):\[ 5 + B = 3\]\[ B = -2 \]Substitute \( A = 5 \) and \( B = -2 \) into \( 2A + B + C = 15 \):\[ 2(5) + (-2) + C = 15 \]\[ 10 - 2 + C = 15 \]\[ C = 7 \]

Step 6: Write the Decomposition

Substitute the values of \( A \), \( B \), and \( C \) into the partial fraction decomposition:\[ \frac{3 x^{2}+15 x+5}{x (x + 1)^2} = \frac{5}{x} + \frac{-2}{x + 1} + \frac{7}{(x + 1)^2} \]So, the partial fraction decomposition is:\[ \frac{5}{x} - \frac{2}{x + 1} + \frac{7}{(x + 1)^2} \]

Key Concepts

Rational Functions

Rational functions are fractions where the numerator and denominator are both polynomials. Understanding these functions helps in various areas of algebra and calculus.
For example, in the given exercise, the rational function is \(\frac{3x^{2}+15x+5}{x^{3}+2x^{2}+x}\).
Grasping how to work with rational functions involves:

• Identifying the degrees of the numerator and denominator.
• Simplifying the fraction by factoring, if possible.

This simplification is the first step used to perform partial fraction decomposition, making complex fractions easier to manage.
One crucial skill in handling rational functions is recognizing common factors to simplify them before decomposition or other operations.

Factoring

Factoring involves breaking down a polynomial into simpler 'factors' that can be multiplied together to give the original polynomial.
In the partial fraction decomposition example, we first factored the denominator \(\frac{3x^{2}+15x+5}{x^{3}+2x^{2}+x}\).
The denominator is factored as follows:

• First, factor out the greatest common factor (GCF). In our case, it's \(x\), resulting in \(x(x^{2}+2x+1)\).
• Next, recognize \(x^{2}+2x+1\) as a perfect square trinomial, simplifying it further to \(x(x+1)^{2}\).

This step is crucial as it lays the groundwork for partial fraction setup by simplifying the expression into terms that can be separated easily.

Solving Equations

The process of solving equations means finding the values that make the equation true. It is essential in every step of partial fraction decomposition.
When equating the numerators after setting up the partial fractions, we get:

• Original equation: \(\frac{3x^{2}+15x+5}{x(x+1)^2} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2}\)
• Combined right-hand denominator: \(\frac{A(x+1)^2 + Bx(x+1) + Cx}{x(x+1)^2}\)
• Equal numerators: \3x^2 + 15x + 5 = A(x^2 + 2x + 1) + Bx^2 + Bx + Cx\.

We solve for constants \A, B,\ and \C\ by matching coefficients with corresponding powers of \x\ in the polynomial. This allows us to break down the decomposed fractions fully.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. Partial fraction decomposition heavily relies on fundamental algebraic procedures.
Key algebraic steps in this process include:

• Factoring polynomials, identifying common terms, and simplifying expressions.
• Setting up and equating expressions to solve for unknowns like constants \A, B,\ and \C\.

For example, we started from \(\frac{3x^{2}+15x+5}{x (x+1)^2}\) and algebraically manipulated it to reach \(\frac{5}{x} - \frac{2}{x+1} + \frac{7}{(x+1)^2}\).
Mastering these algebra skills is essential not only for partial fraction decomposition but also for solving many other types of mathematical problems.