Problem 36 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression is \(3x^{2}-2x+8\).

Step 1: Factorize the Denominator

Firstly, the denominator of the given rational expression, which is \(x^{3}+2 x^{2}+4 x+8\), is factored into simpler forms. But in this case, the polynomial \(x^{3}+2 x^{2}+4 x+8\) doesn't factor into simpler forms using real numbers. Therefore, the partial fraction decomposition is same as the original fraction.

Step 2: Write the Partial Fraction Decomposition

Since the denominator can't be factored into simpler forms, the numerator \(3x^{2}-2x+8\) is the partial fraction decomposition.

Key Concepts

Rational Expressions

Rational expressions are like fractions, but instead of just numbers, they include polynomials in the numerator and the denominator. In mathematical terms, a rational expression is any expression that can be written as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero.
Understanding rational expressions is crucial because they describe numerous real-world relationships, similar to how fractions do with quantities.When working with rational expressions, the goal often includes:

• Simplifying the expression by finding a common factor if possible.
• Performing operations such as addition, subtraction, multiplication, or division.

Rational expressions can become complex, especially when they involve quadratic or cubic polynomials, which need special techniques to manipulate, such as factorization or polynomial division.

Factorization

Factorization is the process of breaking down a polynomial into a product of simpler polynomials or primes. This is often used to simplify expressions or to find their roots. In the context of partial fraction decomposition, factorization helps in splitting a complex rational expression into simpler parts that are easier to work with.
Typically, when we say factorize, we mean finding a set of polynomials, which when multiplied together give the original polynomial. For example, factorizing \( x^2 - 9 \) gives \((x-3)(x+3)\). However, not all polynomials can be neatly factorized using real numbers, such as the cubic polynomial in our problem \(x^3 + 2x^2 + 4x + 8\), which remains irreducible over the reals. This means the polynomial cannot be expressed as a product of linear and/or irreducible quadratic factors with real coefficients.Sometimes, understanding or applying complex numbers might be necessary to factor completely.

Polynomial Long Division

Polynomial long division is a method similar to numerical long division, used when a polynomial needs to be divided by another polynomial, especially when finding partial fraction decompositions. This technique helps divide a higher-degree polynomial by a lower-degree polynomial, producing a quotient and sometimes a remainder.
Here's a simplified process:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the divisor by this result and subtract from the dividend.
• Repeat the process with the new dividend formed by the subtraction.

Polynomial long division is particularly useful when simplifying rational expressions or when tackling irreducible factors where other methods, like straightforward factorization, don't apply. Though it was not employed directly in this problem due to the irreducibility of the denominator, it remains an essential tool in algebra.

Complex Numbers

Complex numbers come into play when dealing with polynomials that do not factor over the real numbers, like in our example. They expand the familiar number line into a plane, allowing solutions to equations that don't have real roots. A complex number has the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).In algebra, complex numbers are crucial for:

• Finding roots of polynomials that are irreducible over the reals.
• Simplifying certain expressions that appear unsolvable with real numbers alone.

Although we didn't need them to decompose the specific rational expression in the exercise, knowing about complex numbers enriches understanding and prepares students for advanced topics in mathematics, such as complex factorization or solving higher-degree polynomial equations.