1 Rational numbers (Section P.1 , page 3)
2 Properties of fractions (Section P.1 , page 13)
3 Factoring polynomials (Section P.4 )
4 Irreducible polynomial (Section P.4 , page 42)
You are probably aware that the nonmedical use of steroids among adolescents and young adults is an ongoing national concern. In fact, many high schools have required students involved in extracurricular activities to submit to random drug testing. Many methods of drug testing are in use, and those who administer the tests are keenly aware of the possibility of what is termed a “false positive,” which occurs when someone who is not a drug user tests positive for using drugs. How likely is it that a student who tests positive actually uses drugs? For a test that is 95% accurate, an expression for the likelihood that a student who tests positive but is not a drug user is
where x is the percent of the student population that uses drugs. This is an example of a rational expression, which we study in this section. In Example 1, we see that in a student population among which only 5% of the students use drugs, the likelihood that a student who tests positive is not a drug user is 50% when the test used is 95% accurate.
1 Define rational expressions.
Recall that the quotient of two integers, ab (b≠0), is a rational number. The quotient of two polynomials is called a rational expression. Following are examples of rational expressions.
We use the same language to describe rational expressions that we use to describe rational numbers. For x2+2x−3x2+5x+6, we call x2+2x−3 the numerator and x2+5x+6 the denominator. The domain of a rational expression is the set of all real numbers except those that result in a zero denominator. For example, the domain of 2x−1 is all real x, x≠1. We write x+6(x−3)(x+4),x≠3, x≠−4 to indicate that 3 and −4 are not in the domain of this rational expression.
With a test that is 95% accurate, an expression for the likelihood that a student who tests positive for drugs is a nonuser is
where x is the percent (in decimal) of the student population that uses drugs. In a student population among which only 5% of the students use drugs, find the likelihood that a student who tests positive is a nonuser.
We convert 5% to a decimal to get x=0.05, and we replace x with 0.05 in the given expression.
So the likelihood that a student who tests positive is a nonuser when the test used is 95% accurate is 50%. Because of the high rate of false positives, a more accurate test is needed.
Rework Example 1 for a student population among which 10% of the students use drugs.
2 Reduce a rational expression to lowest terms.
A rational expression is reduced to its lowest terms or simplified if its numerator and denominator have no common factor other than 1 or −1.
The following property of fractions (see page 13)
is used to simplify rational expressions.
Simplify each expression.
x4+2x3x+2,x≠−2
x2−x−6x3−3x2, x≠0, x≠3
Factor each numerator and denominator and divide out the common factors.
x4+2x3x+2=x3(x+2)x+2=x3(x+2)(x+2)=x3
x2−x−6x3−3x2=(x+2)(x−3)x2(x−3)=(x+2)(x−3)x2(x−3)=x+2x2
Simplify each expression.
2x3+8x23x2+12x
x2−4x2+4x+4
Because x2 is a term, not a factor, in both the numerator and denominator of x2−1x2+2x+1, you cannot remove x2. You can remove only factors common to both the numerator and denominator of a rational expression.
3 Multiply and divide rational expressions.
We use the same rules for multiplying and dividing rational expressions as we do for rational numbers.
In working with the quotient of two rational expressions ABCD=AB⋅DC three polynomials appear as denominators: B and D from AB and CD and C from AB⋅DC.
The best strategy to use when multiplying and dividing rational expressions is to factor each numerator and denominator completely and then divide out the common factors.
Multiply or divide as indicated. Simplify your answer and leave it in factored form.
x2+3x+2x3+3x⋅2x3+6xx2+x−2,x≠0, x≠1, x≠−2
3x2+11x−48x3−40x23x+124x4−20x3, x≠0, x≠5, x≠−4
Factor each numerator and denominator and divide out the common factors.
x2+3x+2x3+3x⋅2x3+6xx2+x−2=(x+2)(x+1)x(x2+3)⋅2x(x2+3)(x−1)(x+2))=(x+2)(x+1)(2x)(x2+3)x(x2+3)(x−1)(x+2)=2(x+1)x−1
3x2+11x−48x3−40x23x+124x4−20x3=3x2+11x−48x3−40x2⋅4x4−20x33x+12=(x+4)(3x−1)8x2(x−5)⋅4x3(x−5)3(x+4)=(x+4)(3x−1)x(4x3)(x−5)8x22(x−5)(3)(x+4)=(3x−1)x6=x(3x−1)6
Multiply or divide as indicated. Simplify your answer.
4 Add and subtract rational expressions.
To add and subtract rational expressions, we use the same rules as for adding and subtracting rational numbers.
Add or subtract as indicated. Simplify your answer and leave both the numerator and denominator in factored form.
x−6(x+1)2+x+8(x+1)2, x≠−1
3x−2x2−5x+6−2x+1x2−5x+6, x≠2, x≠3
x−6(x+1)2+x+8(x+1)2=x−6+x+8(x+1)2=2x+2(x+1)2=2(x+1)(x+1)(x+1)=2x+1
3x−2x2−5x+6−2x+1x2−5x+6=(3x−2)−(2x+1)x2−5x+6=3x−2−2x−1x2−5x+6=x−3(x−2)(x−3)=x−3(x−2)(x−3)=1x−2
Add or subtract as indicated. Simplify your answers.
5x+22x2−36+2(x+10)x2−36, x≠6, x≠−6
4x+1x2+x−12−3x+4x2+x−12, x≠−4, x≠3
When adding or subtracting rational expressions with different denominators, we proceed (as with fractions) by finding a common denominator. The one most convenient to use is called the least common denominator (LCD), and it is the polynomial of least degree that contains each denominator as a factor. In the simplest case, the LCD is the product of the denominators.
Add or subtract as indicated. Simplify your answer and leave it in factored form.
xx+1+2x−1x+3, x≠−1, x≠−3
2xx+1−xx+2, x≠−1, x≠−2
xx+1+2x−1x+3=x(x+3)(x+1)(x+3)+(2x−1)(x+1)(x+1)(x+3)LCD=(x+1)(x+3)=x(x+3)+(2x−1)(x+1)(x+1)(x+3)=x2+3x+2x2+2x−x−1(x+1)(x+3)=3x2+4x−1(x+1)(x+3)
2xx+1−xx+2=2x(x+2)(x+1)(x+2)−x(x+1)(x+1)(x+2)LCD=(x+1)(x+2)=2x(x+2)−x(x+1)(x+1)(x+2)=2x2+4x−x2−x(x+1)(x+2)=x2+3x(x+1)(x +2)=x(x+3)(x+1)(x+2)
Add or subtract as indicated. Simplify your answers.
2xx+2+3xx−5, x≠−2, x≠5
5xx−4−2xx+3, x≠4, x≠−3
Find the LCD for each pair of rational expressions.
x+2x(x−1)2(x+2), 3x+74x2(x+2)3
x+1x2−x−6, 2x−13x2−9
Step 1 The denominators are already completely factored.
Step 2 4x(x−1)(x+2) Product of the different factors
Step 3 LCD=4x2(x−1)2(x+2)3The greatest exponents are 2, 2, and 3.
Step 1 x2−x−6=(x+2)(x−3)x2−9=(x+3)(x−3)
Step 2 (x+2)(x−3)(x+3) Product of the different factors
Step 3 LCD=(x+2)(x−3)(x+3) The greatest exponent on each factor is 1.
Find the LCD for each pair of rational expressions.
x2+3xx2(x+2)2(x−2), 4x2+13x(x−2)2
2x−1x2−25, 3−7x2(x2+4x−5)
When adding or subtracting rational expressions with different denominators, the first step is to find the LCD. Here is the general procedure.
Add or subtract as indicated. Simplify your answer and leave it in factored form.
3x2−1+xx2+2x+1, x≠1, x≠−1
x+2x2−x−3x4(x−1)2, x≠0, x≠1
Step 1 Note that x2−1=(x−1)(x+1) and x2+2x+1=(x+1)2; the LCD is (x+1)2(x−1).
Step 2
Steps 3–4 Add.
Step 1 The LCD is 4x(x−1)2. x2−x=x(x−1)
Step 2
Steps 3–4 Subtract.
Add or subtract as indicated. Simplify your answers.
4x2−4x+4+xx2−4, x≠2, x=−2
2x3(x−5)2−6x2(x2−5x), x≠0, x≠5
5 Identify and simplify complex fractions.
A rational expression that contains another rational expression in its numerator or denominator (or both) is called a complex rational expression or complex fraction. To simplify a complex fraction, we write it as a rational expression in lowest terms.
There are two effective methods of simplifying complex fractions.
Simplify 12+1xx2−42x, x≠0, x≠2, x≠−2, using each of the two methods.
Method 1
Method 2 The LCD of 12, 1x, and x2−42x is 2x.
Simplify: 53x+13x2−253x, x≠0, x≠5, x≠−5
Simplify: x2x−73−12, x≠75
We use Method 1.
Simplify: 5x3x−42−13, x≠45
The least common denominator of two rational expressions is the polynomial of least degree that contains as a factor.
The first step in finding the LCD of two rational expressions is to the denominators completely.
If the denominators of two rational expressions are x2−2x and x2−x−2, the LCD is .
A rational expression that contains another rational expression in its numerator or denominator is called a(n) .
True or False. The expression 12+1x is a complex fraction.
True or False. A polynomial is also a rational expression.
True or False. In simplifying rational expressions to lowest terms, we use a+ba+c=bc.
True or False. The fraction 2x2+3xx+1 is in lowest terms.
In Exercises 9–24, reduce each rational expression to lowest terms. Specify the domain of the rational expression by identifying all real numbers that must be excluded from the domain.
2x+2x2+2x+1
3x−6x2−4x+4
3x+3x2−1
10−5x4−x2
2x−69−x2
15+3xx2−25
2x−11−2x
2−5x5x−2
x2−6x+94x−12
x2−10x+253x−15
7x2+7xx2+2x+1
4x2+12xx2+6x+9
x2−11x+10x2+6x−7
x2+2x−15x2−7x+12
6x4+14x3+4x26x4−10x3−4x2
3x3+x23x4−11x3−4x2
In Exercises 25–42, multiply or divide as indicated. Simplify and leave the numerator and denominator in your answer in factored form. Assume all denominators are nonzero.
x−32x+4⋅10x+205x−15
6x+42x−8⋅x−49x+6
2x+64x−8⋅x2+x−6x2−9
25x2−94−2x⋅4−x210x−6
x2−7xx2−6x−7⋅x2−1x2
x2−9x2−6x+9⋅5x−15x+3
x2−x−6x2+3x+2⋅x2−1x2−9
x2+2x−8x2+x−20⋅x2−16x2+5x+4
2−xx+1⋅x2+3x+2x2−4
3−xx+5⋅x2+8x+15x2−9
x+26÷4x+89
x+320÷4x+129
x2−9x÷2x+65x2
x2−13x÷7x−7x2+x
x2+2x−3x2+8x+16÷x−13x+12
x2+5x+6x2+6x+9÷x2+3x+2x2+7x+12
(x2−9x3+8÷x+3x3+2x2−x−2) 1x2−1
(x2−25x2−3x−4÷x2+3x−10x2−1) x−2x−5
In Exercises 43–60, add and subtract as indicated. Simplify and leave the numerator and denominator in your answer in factored form. Exercises 57–60 require an LCD.
x5+35
74−x4
x2x+1+42x+1
2x7x−3+x7x−3
x2x+1−x2−1x+1
2x+73x+2−x−23x+2
43−x+2xx−3
−21−x+2−xx−1
5xx2+1+2xx2+1
x2(x−1)2+3x2(x−1)2
7x2(x−3)+x2(x−3)
4x4(x+5)2+8x4(x+5)2
xx2−4−2x2−4
5xx2−1−5x2−1
x−22x+1−x2x−1
2x−14x+1−2x4x−1
−xx+2+x−2x−xx−2
3xx−1+x+1x−2xx+1
In Exercises 61–68, find the LCD for each pair of rational expressions.
53x−6, 2x4x−8
5x+17+21x, 1−x3+9x
3−x4x2−1, 7x(2x+1)2
14x(3x−1)2, 2x+79x2−1
1−xx2+3x+2, 3x+12x2−1
5x+9x2−x−6, x+5x2−9
7−4xx2−5x+4, x2−xx2+x−2
13xx2−2x−3, 2x2−4x2+3x+2
In Exercises 69–84, perform the indicated operations and simplify the result. Leave the numerator and denominator in your answer in factored form.
5x−3+2xx2−9
3xx−1+xx2−1
2xx2−4−xx+2
3x−1x2−16−2x+1x−4
x−2x2+3x−10+x+3x2+x−6
x+3x2−x−2+x−1x2+2x+1
2x−39x2−1+4x−1(3x−1)2
3x+1(2x+1)2+x+34x2−1
x−3x2−25−x−3x2+9x+20
2xx2−16−2x−7x2−7x+12
3x2−4+12−x−12+x
25+x+5x2−25+75−x
x+3ax−5a−x+5ax−3a
3x−a2x−a−2x+a3x+a
1x+h−1x
1(x+h)2−1x2
In Exercises 85–98, perform the indicated operations and simplify the result. Leave the numerator and denominator in your answer in factored form.
2x3x2
−6x2x3
1x1−1x
1x21x2−1
1x−11x+1
2−2x1+2x
1x−x1−1x2
x−1x1x2−1
x−xx+12
x+xx+12
1x+h−1xh
1(x+h)2−1x2h
1x−a+1x+a1x−a−1x+a
1x−a+1x+axx−a−ax+a
Bearing length. The length (in centimeters) of a bearing in the shape of a cylinder is given by 125.6πr2, where r is the radius of the base of the cylinder. Find the length of a bearing with a diameter of 4 centimeters, using 3.14 as an approximation of π.
Toy box height. The height (in feet) of an open toy box that is twice as long as it is wide is given by 13.5−2x26x, where x is the width of the box. Find the height of a toy box that is 1.5 feet wide.
Diluting a mixture. A 100-gallon mixture of citrus extract and water is 3% citrus extract.
Write a rational expression in x whose values give the percentage (in decimal form) of the mixture that is citrus extract when x gallons of water are added to the mixture.
Find the percentage of citrus extract in the mixture assuming that 50 gallons of water are added to it.
Acidity in a reservoir. A half-full 400,000-gallon reservoir is found to be 0.75% acid.
Write a rational expression in x whose values give the percentage (in decimal form) of acid in the reservoir when x gallons of water are added to it.
Find the percentage of acid in the reservoir assuming that 100,000 gallons of water are added to it.
Diet lemonade. A company packages its powdered diet lemonade mix in containers in the shape of a cylinder. The top and bottom are made of a tin product that costs 5 cents per square inch. The side of the container is made of a cheaper material that costs 1 cent per square inch. The height of any cylinder can be found by dividing its volume by the area of its base.
Assuming that x is the radius of the base (in inches), write a rational expression in x whose values give the cost of each container, assuming that the capacity of the container is 120 cubic inches.
Find the cost of a container whose base is 4 inches in diameter, using 3.14 as an approximation of π.
Storage containers. A company makes storage containers in the shape of an open box with a square base. All five sides of the box are made of a plastic that costs 40 cents per square foot. The height of any box can be found by dividing its volume by the area of its base.
Assuming that x is the length of the base (in feet), write a rational expression in x whose values give the cost of each container, assuming that the capacity of the container is 2.25 cubic feet.
Find the cost of a container whose base is 1.5 feet long.
In Exercises 105–114, simplify each expression.
(3x+4)2−(2x+3)2(3x+4)2
(x2+1)−x(2x)(x2+1)2
(x2−2x+1)(2x+2)−(x2+2x+1)(2x−2)(x2−2x+1)2
(3x2+4x+1)(4x+1)−(2x2+x−1)(6x+4)(3x2+4x+1)2
x2−x−20x2−25⋅x2−x−2x2+2x−8÷x+1x2+5x
y4−x4y2−2xy+x2÷(y2x+x3y3−x3⋅y4+y2x2+x4y2x−yx2+x3)
1a−1a+1a−1a+1a−1a
1+x−yx+y1−x−yx+y÷1+x2−y2x2+y21−x2−y2x2+y2
(x21−x4+2x41−x8)÷x2+1x
x2+y2y−x1y−1x⋅x2−y2x3+y3
In Exercises 115–120, state whether the statement is True or False.
2√3+5√3=(2+5)√3=7√3
x√2+y√2−z√2=(x+y−z)√2
√(−4)2=−4
(√4)2=4
√4⋅16=√4⋅√16
(a+b)2=a2+b2