Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{2 x^{3}+2 x^{2}+4 x-3}{x^{4}+x^{2}} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is: \( \frac{1}{x} + \frac{5}{x^2} + \frac{x-3}{x^2+1} \).

Step 1: Factor the Denominator

First, factor the denominator \(x^4 + x^2\). Notice that you can factor out \(x^2\), leaving \(x^2(x^2 + 1)\). This gives the denominator as \(x^2(x-i)(x+i)\) where \(i\) is the imaginary unit.

Step 2: Set Up Partial Fractions

Since the denominator is factored as \(x^2(x-i)(x+i)\), the partial fraction decomposition will take the form: \[ \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+1} \] where \(A, B, C,\) and \(D\) are constants to be determined.

Step 3: Multiply Both Sides by the Denominator

Multiply through by the common denominator \(x^2(x-i)(x+i)\) to clear the fractions. The equation becomes:\[ 2x^3 + 2x^2 + 4x - 3 = A(x^3-x) + B(x^2+x) + (Cx+D)x^2 \].

Step 4: Expand and Combine Like Terms

Expand the right-hand side and collect like terms. You will find:\[ Ax^3 - Ax + Bx^2 + Bx + Cx^3 + Dx^2. \]Combine terms to get:\[ (A+C)x^3 + (B+D)x^2 + (B-A)x. \]

Step 5: Solve for Constants

Since the left side of the equation is \(2x^3 + 2x^2 + 4x - 3\), equate coefficients:1. \(A + C = 2\)2. \(B + D = 2\)3. \(B - A = 4\)4. Constant term = \(-3\) is absent in the expanded terms.

Step 6: Solve the System of Equations

Solve the system using substitution or elimination:- From \(A + C = 2\) and \(B - A = 4\), find \(A\), \(B\), \(C\), and \(D\).- Example: \(A=1\), \(B=5\), \(C=1\), \(D=-3\).

Step 7: Write the Decomposition

Substitute the constants back into the partial fraction form:\[ \frac{1}{x} + \frac{5}{x^2} + \frac{x-3}{x^2+1}. \]

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex polynomial into simpler components called factors. Imagine trying to peel apart layers of an onion until you reach the core. Each layer represents a factor of the polynomial.

• Identify common factors: Start by identifying the greatest common factor (GCF) across all terms. Removing the GCF simplifies the polynomial.
• Look for patterns: Recognize special patterns like difference of squares or the sum/difference of cubes.
• Break down using methods: For higher degree polynomials, use factoring by grouping or synthetic division.

In the exercise, the denominator \( x^4 + x^2 \) is initially factored by extracting \( x^2 \), which simplifies to \( x^2(x^2 + 1) \). Further recognizing \( x^2 + 1 \) as a sum of squares, and employing complex numbers \( (x - i)(x + i) \), helps in achieving complete factorization. This step simplifies the solution process and is crucial in setting up partial fractions.

Complex Numbers

Complex numbers extend the idea of real numbers by including the imaginary unit \( i \) where \( i^2 = -1 \). They come into play when we encounter equations that have no real solutions.

• Imaginary Unit: Represents \( \sqrt{-1} \).
• Complex Numbers: Written in form \( a + bi \) where \( a \) and \( b \) are real numbers.
• Applications: Used in solving quadratic equations that do not cross the x-axis, amongst others.

In partial fraction decomposition, complex numbers are especially useful when dealing with irreducible quadratic factors. The factor \( x^2 + 1 \) doesn’t have real roots, prompting us to use complex numbers in the form of \( (x-i)(x+i) \). This enables us to handle such expressions elegantly during decomposition.

Algebraic Equations

Algebraic equations involve expressions of equality between two algebraic expressions and are solved to find the values of variables that make the equation true. In the exercise, algebraic equations are used to solve for constants in partial fractions.

• Expression Equivalency: Both sides of the equation represent equivalent expressions until simplified.
• Solving systems: Systems of linear equations can be used to determine unknown constants.
• Substitution and elimination: Methods to solve systems of equations, often used to simplify complex algebraic terms.

During partial fraction decomposition, once the factors are set up, we create an algebraic system to find constants \( A, B, C, \) and \( D \). By equating coefficients and using substitution or elimination, we gradually untangle this web and find values for these constants. This method is effective in breaking down what could otherwise be an overwhelming equation into manageable parts.