Problem 45 - Precalculus with Limits

Question

Improper Rational Expression Decomposition, write the partial fraction decomposition of the improper rational expression. $$\frac{2 x^{3}-x^{2}+x+5}{x^{2}+3 x+2}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given improper rational expression is \(2x - 3 + \frac{5}{x + 1}+ \frac{2}{x + 2}\)

Step 1: Factor the Denominator

First, factorize the denominator \(x^{2} + 3x + 2\), thus i.e \(x^{2} + 3x + 2 = (x + 1)(x + 2)\).

Step 2: Polynomial Division

Next, perform the polynomial division of \(2x^{3} - x^{2} + x + 5\) by \((x + 1)(x + 2)\) to get the expression \(2x - 3 + \frac{7}{(x + 1)(x + 2)}\). This is because the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator making it an improper fraction.

Step 3: Decompose into Partial Fractions

Break down the fraction \(\frac{7}{(x + 1)(x + 2)}\) into partial fractions. This gives us, \(\frac{7}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2}\). Solving for A and B, it gives A = 5 and B =2

Key Concepts

Improper Rational Expressions

Improper rational expressions arise when the degree of the numerator is greater than or equal to the degree of the denominator. In the expression \(\frac{2x^3-x^2+x+5}{x^2+3x+2}\), the numerator has a degree of 3, while the denominator has a degree of 2, making it an improper rational expression.
Understanding and identifying these expressions is essential because they often need to be simplified before further manipulation, such as creating partial fraction decompositions.

To tackle these expressions, it's crucial to first simplify them using polynomial division. This turns the improper fraction into a proper one, removing unnecessary complexity from the problem at hand.

Polynomial Division

Polynomial division is a crucial tool for simplifying improper rational expressions. It's a process similar to long division, allowing us to divide polynomials systematically.
For instance, in the expression \(2x^3 - x^2 + x + 5\) divided by \((x^2 + 3x + 2)\), we'll break down the polynomial into simpler components. This will result in both a quotient and a remainder.

• The quotient is the result of the division, while the remainder is the leftover part that cannot be evenly divided by the polynomial.
• For our example, the result is \(2x - 3 + \frac{7}{(x+1)(x+2)}\).

This result refines the original expression into a form more suitable for partial fraction decomposition, ensuring that the fractions can be properly separated and analyzed further.

Factoring Polynomials

Factoring polynomials simplifies expressions significantly and opens the door for various other methods like partial fraction decomposition.
In our initial expression's denominator, \(x^2 + 3x + 2\) can be factored to \((x+1)(x+2)\). Recognizing patterns such as these not only simplifies the work required but also helps avoid mistakes in further calculations.

When factoring polynomials, always:

• Look for common patterns like quadratic trinomials, differences of squares, or sums/differences of cubes.
• Practice makes recognizing these patterns quicker and more intuitive.
• Remember that accurate factoring is critical for the success of any further decomposition or simplification attempted on the expression.

Polynomials do not always factor easily, but persistence in identifying the simplest terms and checking your work will lead to more accurate math solutions.