Problem 44 - Precalculus with Limits

Question

Improper Rational Expression Decomposition, write the partial fraction decomposition of the improper rational expression. $$\frac{x^{2}-4 x}{x^{2}+x+6}$$

Step-by-Step Solution

Answer

The Partial Fraction Decomposition of the given improper rational expression \(\frac{x^{2}-4x}{x^{2}+x+6}\) is \(A/(x^2 + x + 6)\) where A can be found by equating coefficients.

Step 1: Perform Polynomial Division

First, let's perform polynomial division since the rational fraction is an improper fraction, using this division we can rewrite the fraction as a sum of a polynomial and a proper fraction.

Step 2: Factorize the denominator

Next is to factorize the denominator of the rational function, which \(x^2 + x + 6\), but this function cannot be factored any further as it has complex roots.

Step 3: Rewrite as a Partial Fraction Division

Now, rewrite the expression as a Partial Fraction Decomposition considering the factors in the denominator. As it is already in its minimal form, rewrite it as \(A/(x^2 + x + 6)\)

Step 4: Equate Coefficients

Equate coefficients from both side of equation allows us to find the constant A, which is part of the Partial Fraction Decomposition.

Key Concepts

Improper Rational Expression

An improper rational expression is a fraction where the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. This is contrary to a proper fraction where the denominator's degree is larger than the numerator's degree.
For instance, in the expression \( \frac{x^2 - 4x}{x^2 + x + 6} \), both the numerator and the denominator are of degree 2. Thus, it is an improper rational expression.

To handle these expressions more easily, we often use partial fraction decomposition. However, before proceeding to the decomposition for improper fractions, it's ideal to perform polynomial division first.

• This step is crucial as it simplifies the expression into a polynomial plus a proper fraction (where the numerator's degree is less than the denominator's degree).
• The result is an easier expression to manipulate, particularly when integrating or solving other problems with rational expressions.

Polynomial Division

Polynomial division is very similar to long division with numbers and comes in handy when dealing with improper rational expressions. This process involves dividing the polynomial in the numerator by the polynomial in the denominator, effectively simplifying complex fractions.
In our given exercise, you will divide \(x^2 - 4x\) by \(x^2 + x + 6\). Although it might seem challenging at first, breaking it down into steps helps.

• Start by dividing the first term of the numerator by the first term of the denominator.
• Multiply the entire denominator by the quotient found in the first step.
• Subtract this result from the original numerator.
• Repeat this process with the result until the degree of the remainder is less than the degree of the denominator.

Through this, you obtain a quotient and a remainder. The remainder forms part of the proper fraction you need for partial fraction decomposition.

Ultimately, this simplifies our workflow when dealing with other processes like integration or equating coefficients.

Complex Roots

Complex roots often appear when trying to factor polynomials, and they occur when the equation cannot be simplified using real numbers alone. This typically arises with quadratic equations which do not factor neatly over the set of real numbers.
In the denominator \(x^2 + x + 6\) from our exercise, you might try to factor it to make partial fraction decomposition easier. However, because it has complex roots, you cannot simplify it further by factorizing into real linear factors.

• You find complex roots by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Here, \(a = 1\), \(b = 1\), and \(c = 6\), resulting in a negative discriminant \(b^2 - 4ac\), indicating complex roots.

Recognizing complex roots prevents unnecessary attempts at factoring using other methods. Instead, you leave it as is for the partial fraction decomposition.

Understanding their presence also helps when solving equations, as it tells us the function does not cross the x-axis, among other insights.