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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 P ( x ) Q ( x ) 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:

2 x −3 + −1 x + 2

We would first need to find a common denominator, ( x + 2 ) ( x −3 ) .

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

2 x 3 ( x + 2 x + 2 ) + 1 x + 2 ( x 3 x 3 ) =                        2 x + 4 x + 3 ( x + 2 ) ( x 3 ) = x + 7 x 2 x 6

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.

x + 7 x 2 x −6 Simplified sum = 2 x −3 + −1 x + 2 Partial fraction decomposition

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 x 2 x −6 are ( x −3 ) ( x + 2 ) , 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 P ( x ) Q ( x ) : Q ( x ) Has nonrepeated linear factors

The partial fraction decomposition    of P ( x ) Q ( x ) when Q ( x ) has nonrepeated linear factors and the degree of P ( x ) is less than the degree of Q ( x ) is

P ( x ) Q ( x ) = A 1 ( a 1 x + b 1 ) + A 2 ( a 2 x + b 2 ) + A 3 ( a 3 x + b 3 ) + + A n ( a n x + b n ) .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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