Problem 41 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}+3 x+14}{x^{3}-8}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{4 x^{2}+3 x+14}{x^{3}-8}\) is \(\frac{A}{x-2}+\frac{Bx+C}{x^{2}+2x+4}\), where \(A\), \(B\) and \(C\) are constants that can be found by equating coefficients from Step 3.

Step 1: Factorize the denominator

First, factorize the polynomial in the denominator. \(x^{3}-8\) can be factored as \((x-2)(x^{2}+2x+4)\).

Step 2: Decompose into partial fractions

Now, decompose the fraction \(\frac{4 x^{2}+3 x+14}{(x-2)(x^{2}+2x+4)}\) into simpler fractions. Let's write it as \( \frac{4 x^{2}+3 x+14}{(x-2)(x^{2}+2x+4)} = \frac{A}{x-2}+\frac{Bx+C}{x^{2}+2x+4}\).

Step 3: Solve for A, B and C

Get rid of the fraction by multiplying throughout by the denominator to get \(4 x^{2}+3 x+14 = A(x^{2}+2x+4) + (Bx+C)(x-2)\). Expand and collect like terms. Then, equate the coefficients on both sides to solve for \(A\), \(B\) and \(C\).

Key Concepts

Polynomial Factorization

Understanding polynomial factorization is crucial when working with partial fraction decomposition. Factorizing a polynomial is similar to breaking down a number into its prime factors. This process involves expressing the polynomial as a product of its irreducible factors, often involving smaller degree polynomials or constants.

For the exercise \( \frac{4x^{2}+3x+14}{x^{3}-8} \), the denominator \( x^{3} - 8 \) is a difference of cubes and factorizes into \( (x - 2)(x^{2} + 2x + 4) \). This is a key step because partial fraction decomposition requires the denominator to be written as a product of its factors before the expression can be split into simpler fractions. It's important to recognize different polynomial forms, like difference of squares or cubes, to factorize efficiently.

Rational Expressions

In algebra, a rational expression is one that involves ratios of polynomials. These expressions can often seem complicated, but with the right approach, they can be simplified to make the handling of equations much easier. Partial fraction decomposition is one of the techniques used to simplify such expressions, especially when integrating or finding the inverse Laplace transform.

When you see an expression like \( \frac{4x^{2}+3x+14}{x^{3}-8} \), it represents a complex relationship that isn't always straightforward to work with, particularly in calculus. Decomposing the fraction into simpler parts allows for easier manipulation and solution of equations. It's like breaking down a complex machine into smaller, more understandable components.

Algebraic Fractions

At their core, algebraic fractions are fractions where both the numerator and the denominator are polynomials. Much like the fractions we learn in basic arithmetic, algebraic fractions follow similar rules but require more steps and knowledge of algebra to simplify, add, subtract, multiply, or divide.

In the context of partial fraction decomposition, these algebraic fractions are expressed as a sum of simpler fractions, allowing for the ease of operations like integration. For instance, \( \frac{4x^{2}+3x+14}{x^{3}-8} \), once decomposed, gives more manageable expressions that can be dealt with individually. Being familiar with algebraic fractions is key to understanding and mastering more advanced mathematics.