11.4 Partial fractions

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In this section, you will:
• Decompose   P( x )/Q( x ) ,  where  Q( x )  has only nonrepeated linear factors.
• Decompose   P( x )/Q( x ) ,  where  Q( x )  has repeated linear factors.
• Decompose   P( x )/Q( x ) ,  where  Q( x )  has a nonrepeated irreducible quadratic factor.
• Decompose   P( x )/Q( x ) ,  where  Q( x )  has a repeated irreducible quadratic factor.

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.

Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.

Decomposing $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ Where Q(x) Has only nonrepeated linear factors

Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at partial fraction decomposition    , which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified rational expression    to the original expressions, called the partial fractions    .

For example, suppose we add the following fractions:

$\frac{2}{x-3}+\frac{-1}{x+2}$

We would first need to find a common denominator, $\text{\hspace{0.17em}}\left(x+2\right)\left(x-3\right).$

Next, we would write each expression with this common denominator and find the sum of the terms.

Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.

$\underset{\begin{array}{l}\\ \text{Simplified}\text{\hspace{0.17em}}\text{sum}\end{array}}{\frac{x+7}{{x}^{2}-x-6}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\underset{\begin{array}{l}\\ \text{Partial}\text{\hspace{0.17em}}\text{fraction}\text{\hspace{0.17em}}\text{decomposition}\end{array}}{\frac{2}{x-3}+\frac{-1}{x+2}}$

We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.

When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of $\text{\hspace{0.17em}}{x}^{2}-x-6\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}\left(x-3\right)\left(x+2\right),\text{\hspace{0.17em}}$ the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.

Partial fraction decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}:Q\left(x\right)\text{\hspace{0.17em}}$ Has nonrepeated linear factors

The partial fraction decomposition    of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}Q\left(x\right)\text{\hspace{0.17em}}$ has nonrepeated linear factors and the degree of $\text{\hspace{0.17em}}P\left(x\right)\text{\hspace{0.17em}}$ is less than the degree of $\text{\hspace{0.17em}}Q\left(x\right)\text{\hspace{0.17em}}$ is

$\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{\left({a}_{1}x+{b}_{1}\right)}+\frac{{A}_{2}}{\left({a}_{2}x+{b}_{2}\right)}+\frac{{A}_{3}}{\left({a}_{3}x+{b}_{3}\right)}+\cdot \cdot \cdot +\frac{{A}_{n}}{\left({a}_{n}x+{b}_{n}\right)}.$

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