Problem 19 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{1}{2 x^{2}+x}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{1}{2x^{2} + x}\) = \(\frac{1}{x} + \frac{2}{2x + 1}\).

Step 1: Factor the Denominator

Factor the denominator of the given rational function. The denominator is \(2x^{2} + x\). Factoring it out, you obtain \(x(2x + 1)\).

Step 2: Write the Decomposition Form

Since the denominator factors out to \(x(2x + 1)\), the decomposition can be written as \(\frac{1}{x(2x + 1)} = \frac{A}{x} + \frac{B}{2x + 1}\) where A and B are constants that need to be determined.

Step 3: Clearing the Denominator

Multiply through by the denominator, \(x(2x + 1)\), to clear the fractions. It results in \(1 = A(2x + 1) + Bx\).

Step 4: Solve for the Constants

Set up two equations by substituting suitable values for x in \(1 = A(2x + 1) + Bx\). By setting x=0, obtain A=1. By setting x=-1/2, obtain B=2.

Step 5: Write the Partial Fraction Decomposition

Substitute A and B into the decomposition. Hence, the partial fraction decomposition of \(\frac{1}{2x^{2} + x}\) = \(\frac{1}{x} + \frac{2}{2x + 1}\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of integers or real numbers in the numerator and denominator, we have polynomials. These can often be complex and difficult to work with, especially when you need to add, subtract, multiply, or divide them.

In the context of algebra, a rational expression like \(\frac{1}{2x^2 + x}\) can be simplified or manipulated for various purposes such as solving equations or calculus operations like integration. One common technique used for simplifying rational expressions is partial fraction decomposition. It's particularly useful when integrating rational functions or finding limits in calculus.

Factoring Polynomials

Factoring polynomials is an essential skill when working with algebraic fractions and rational expressions. It's the process of breaking down a complex polynomial into simpler components, or 'factors', that can multiply together to give the original polynomial.

For instance, the polynomial \(2x^2 + x\) from our exercise can be factored into \(x(2x + 1)\). Factoring makes it easier to simplify and perform operations on polynomials. When you factor a polynomial, you're looking for either common factors or patterns like the difference of squares, perfect square trinomials, or other factorable patterns used in advanced algebra. To improve your factoring skills, practice identifying the greatest common factor (GCF), and recognize the various patterns that polynomials might have to facilitate the factoring process.

Algebraic Fractions

Algebraic fractions are simply fractions with polynomials in the numerator, denominator, or both. They can seem daunting at first, but the principles used to handle them are extensions of those we use with ordinary fractions.

Like with any fraction, you can add, subtract, multiply, and divide algebraic fractions by finding common denominators, cross-multiplying, etc. But due to the presence of variables, algebraic fractions allow you to carry out more than just arithmetic operations; you can simplify, expand, and factor them too. When you come across complex algebraic fractions, partial fraction decomposition becomes a valuable tool. It decomposes the fraction into a sum or difference of simpler rational expressions, making it significantly easier to work with them in various math problems.