Section 12.2 Multiplication and Division of Rational Expressions
ΒΆObjectives: PCC Course Content and Outcome Guide
permalinkIn the last section, we learned some rational function applications. In this section, we will learn how to simplify rational expressions, and how to multiply and divide them.
Subsection 12.2.1 Simplifying Rational Expressions
permalinkConsider the two rational functions below. At first glance, which function looks simpler?
permalinkIt can be argued that the function g is simpler, at least with regard to the ease with which we can determine its domain, quickly evaluate it, and also determine where its function value is zero. All of these things are considerably more difficult with the function f.
permalinkThese two functions are actually the same function. Using factoring and the same process of canceling that's used with numerical ratios, we will learn how to simplify the function f into the function g. (The full process for simplifying f(x)=8x3β12x2+8xβ122x3β3x2+10xβ15 will be shown in Example 12.2.8.)
permalinkTo see a simple example of the process for simplifying a rational function or expression, let's look at simplifying 1421 and (x+2)(x+7)(x+3)(x+7) by canceling common factors:
permalinkThe statement βfor xβ β7β was added when the factors of x+7 were canceled. This is because (x+2)(x+7)(x+3)(x+7) was undefined for x=β7, so the simplified version must also be undefined for x=β7.
Warning 12.2.2. Cancel Factors, not Terms.
It may be tempting to want to try to simplify x+2x+3 into 23 by canceling each x that appears. But these x's are terms (pieces that are added with other pieces), not factors. Canceling (an act of division) is only possible with factors (an act of multiplication).
permalinkThe process of canceling factors is key to simplifying rational expressions. If the expression is not given in factored form, then this will be our first step. We'll now look at a few more examples.
Example 12.2.3.
Simplify the rational function formula Q(x)=3xβ12x2+xβ20 and state the domain of Q.
To start, we'll factor the numerator and denominator. We'll then cancel any factors common to both the numerator and denominator.
The domain of this function will incorporate the explicit domain restriction \(x\neq 4\) that was stated when the factor of \(x-4\) was canceled from both the numerator and denominator. We will also exclude \(-5\) from the domain as this value would make the denominator zero. Thus the domain of \(Q\) is \(\left\{x\mid x\neq -5,4\right\}\text{.}\)
Warning 12.2.4.
When simplifying the function Q in Example 12.2.3, we cannot simply write Q(x)=3x+5. The reason is that this would result in our simplified version of the function Q having a different domain than the original Q. More specifically, for our original function Q it held that Q(4) was undefined, and this still needs to be true for the simplified form of Q.
Example 12.2.5.
Simplify the rational function formula R(y)=βyβ2y22y3βy2βy and state the domain of R.
The domain of this function will incorporate the explicit restrictions \(y\neq 0, y\neq -\frac{1}{2}\) that were stated when the factors of \(y\) and \(2y+1\) were canceled from both the numerator and denominator. Since the factor \(y-1\) is still in the denominator, we also need the restriction that \(y\neq 1\text{.}\) Therefore the domain of \(R\) is \(\left\{y\mid y\neq -\frac{1}{2},0,1\right\}\text{.}\)
Example 12.2.6.
Simplify the expression 9y+2y2β5y2β25.
To start, we need to recognize that \(9y+2y^2-5\) is not written in standard form (where terms are written from highest degree to lowest degree). Before attempting to factor this expression, we'll re-write it as \(2y^2+9y-5\text{.}\)
Example 12.2.7.
Simplify the expression β48z+24z2β3z34βz.
To begin simplifying this expression, we will rewrite each polynomial in descending order. Then we'll factor out the GCF, including the constant \(-1\) from both the numerator and denominator because their leading terms are negative.
Example 12.2.8.
Simplify the rational function formula f(x)=8x3β12x2+8xβ122x3β3x2+10xβ15 and state the domain of f.
To simplify this rational function, we'll first note that both the numerator and denominator have four terms. To factor them we'll need to use factoring by grouping. (Note that if this technique didn't work, very few other approaches would be possible.) Once we've used factoring by grouping, we'll cancel any factors common to both the numerator and denominator and state the associated restrictions.
In determining the domain of this function, we'll need to account for any implicit and explicit restrictions. When the factor \(2x-3\) was canceled, the explicit statement of \(x\neq \frac{3}{2}\) was given. The denominator in the final simplified form of this function has \(x^2+5\text{.}\) There is no value of \(x\) for which \(x^2+5=0\text{,}\) so the only restriction is that \(x\neq \frac{3}{2}\text{.}\) Therefore the domain is \(\left\{ x\mid x\neq \frac{3}{2}\right\}\text{.}\)
Example 12.2.9.
Simplify the expression 3yβxx2βxyβ6y2. In this example, there are two variables. It is still possible that in examples like this, there can be domain restrictions when simplifying rational expressions. However since we are not studying functions of more than one variable, this textbook ignores domain restrictions with examples like this one.
Subsection 12.2.2 Multiplication of Rational Functions and Expressions
permalinkRecall the property for multiplying fractions A.2.3.16, which states that the product of two fractions is equal to the product of their numerators divided by the product of their denominators. We will use this same property for multiplying rational expressions.
When multiplying fractions, one approach is to multiply the numerator and denominator, and then simplify the fraction that results by determining the greatest common factor in both the numerator and denominator, like this:
permalinkThis approach works great when we can easily identify that 6 is the greatest common factor in both 42 and 90. But in more complicated instances, it isn't always an easy approach. It also won't work particularly well when we have (x+2) instead of 2 as a factor, as we'll see shortly.
Another approach to multiplying and simplifying fractions involves utilizing the prime factorization of each the numerator and denominator, like this:
permalinkThe method for multiplying and simplifying rational expressions is nearly identical, as shown here:
permalinkThis process will be used for both multiplying and dividing rational expressions. The main distinctions in various examples will be in the factoring methods required.
Example 12.2.10.
Multiply the rational expressions: x2β4xx2β4β 4β4x+x220βxβx2.
Note that to factor the second rational expression, we'll want to re-write the terms in descending order for both the numerator and denominator. In the denominator, we'll first factor out \(-1\) as the leading term is \(-x^2\text{.}\)
Example 12.2.11.
Multiply the rational expressions: p2q43rβ 9r2pq2. Note this book ignores domain restrictions on multivariable expressions.
We won't need to factor anything in this example, and can simply multiply across and then simplify.
Subsection 12.2.3 Division of Rational Functions and Expressions
permalinkWe can divide rational expressions using the property for dividing fractions A.2.4.18, which simply requires that we change dividing by an expression to multiplying by its reciprocal. Let's look at a few examples.
Example 12.2.12.
Divide the rational expressions: x+2x+5Γ·x+2xβ3.
Remark 12.2.13.
In the first step of 12.2.12, the restriction xβ 3 was used. We hadn't canceled anything yet, so why is there this restriction already? It's because the original expression x+2x+5Γ·x+2xβ3 had xβ3 in a denominator, which means that 3 is not a valid input. In the first step of simplifying, the xβ3 denominator went to the numerator and we lost the information that 3 was not a valid input, so we stated it explicitly. Always be sure to compare the restrictions of the original expression with each step throughout the process.
Example 12.2.14.
Simplify the rational expression using division: 3xβ62x+10x2β43x+15.
To begin, we'll note that the larger fraction bar is denoting division, so we will use multiplication by the reciprocal. After that, we'll factor each expression and cancel any common factors.
Example 12.2.15.
Divide the rational expressions: x2β5xβ14x2+7x+10Γ·xβ7x+4.
Example 12.2.16.
Divide the rational expressions: (p4β16)Γ·p4β2p32p.
Note here that we didn't have to include a restriction in the very first step. That restriction would have been \(p\neq0\text{,}\) but since \(0\) still cannot be inputted into any of the subsequent expressions, we don't need to explicitly state \(p\neq0\) as a restriction because the expressions tell us that implicitly already.
Example 12.2.17.
Divide the rational expressions: 3x2x2β9y2Γ·6x3x2β2xyβ15y2. Note this book ignores domain restrictions on multivariable expressions.
Example 12.2.18.
Divide the rational expressions: m2n2β3mnβ42mnΓ·(m2n2β16). Note this book ignores domain restrictions on multivariable expressions.
Reading Questions 12.2.4 Reading Questions
1.
What is the difference between a factor and a term?
2.
When canceling pieces of rational function expression to simplify it, what kinds of pieces are the only acceptable pieces to cancel?
3.
When you simplify a rational function expression, you may need to make note of a .
Exercises 12.2.5 Exercises
Review and Warmup
9.
Factor the given polynomial.
t2β36=
10.
Factor the given polynomial.
x2β4=
11.
Factor the given polynomial.
x2+12x+32=
12.
Factor the given polynomial.
y2+13y+40=
13.
Factor the given polynomial.
y2β3y+2=
14.
Factor the given polynomial.
r2β15r+56=
15.
Factor the given polynomial.
3r2β15r+18=
16.
Factor the given polynomial.
10t2β30t+20=
17.
Factor the given polynomial.
2t10+10t9+12t8=
18.
Factor the given polynomial.
6t5+18t4+12t3=
19.
Factor the given polynomial.
144x2β24x+1=
20.
Factor the given polynomial.
81x2β18x+1=
Simplifying Rational Expressions with One Variable
21.
Simplify the following expressions, and if applicable, write the restricted domain on the simplified expression.
y+4y+4=
y+44+y=
yβ4yβ4=
yβ44βy=
22.
Simplify the following expressions, and if applicable, write the restricted domain on the simplified expression.
y+10y+10=
y+1010+y=
yβ10yβ10=
yβ1010βy=
23.
Select all correct simplifications, ignoring possible domain restrictions.
x+6x+7=67
7x+6x+6=7
6x+6=1x
x+6x=6
7x+67=x+6
6x+6=1x+1
x7x=17
6xx=6
x+6x+6=1
x+66=x
7(xβ6)xβ6=7
24.
Select all correct simplifications, ignoring possible domain restrictions.
x+77=x
7xx=7
x+7x+4=74
4x+74=x+7
x+7x+7=1
x+7x=7
4x+7x+7=4
4(xβ7)xβ7=4
7x+7=1x
x4x=14
7x+7=1x+1
25.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
tβ10(tβ4)(tβ10)=
26.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
t+7(tβ10)(t+7)=
27.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
3(tβ3)(tβ8)(tβ3)=
28.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
β8(xβ9)(xβ6)(xβ9)=
29.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
(x+6)(xβ2)2βx=
30.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
(yβ3)(yβ9)9βy=
31.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
9yβ63yβ7=
32.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
β6r+30rβ5=
33.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
β2rr2+3r=
34.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
9tt2+8t=
35.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
3tβt2t2β9t+18=
36.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
tβt2t2β6t+5=
37.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
x2+5x25βx2=
38.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
x2β3x9βx2=
39.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
βy2+y3β2yβy2=
40.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
βy2+5y5+4yβy2=
41.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
3r2+5r+2βr+4β5r2=
42.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
5r2+8r+3βr+5β6r2=
43.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
r2+6r+8β4rβr2β4=
44.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
t2βtβ2β2tβt2β1=
45.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
βt2β11tβ30t2β25=
46.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
βx2β7xβ12x2β9=
47.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
2x2βxβ3β11xβ5β6x2=
48.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
5y2+11y+6β11yβ5β6y2=
49.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
4y3βy4y2β2yβ8=
50.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
β2r2βr3r2β4=
51.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
r6β3r5β18r4r6β11r5+30r4=
52.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
r5+3r4β4r3r5+2r4β3r3=
53.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
t3+8t2β4=
54.
Simplify the following expression, and if applicable, write the restricted domain on the simplified expression.
t3β125t2β25=
Simplifying Rational Expressions with More Than One Variable
55.
Simplify this expression.
5xyβx2y2x2y2+xyβ30=
56.
Simplify this expression.
5xrβx2r2x2r2βxrβ20=
57.
Simplify this expression.
4y+16ty2+5yt+4t2=
58.
Simplify this expression.
2y+10ty2+8yt+15t2=
59.
Simplify this expression.
βr2+rx+12x2r2β16x2=
60.
Simplify this expression.
βr2βrt+12t2r2β9t2=
61.
Simplify this expression.
2r2y2+5ry+3β11ryβ5β6r2y2=
62.
Simplify this expression.
3t2x2+5tx+2β7txβ2β5t2x2=
Simplifying Rational Functions
63.
Simplify the function formula, and if applicable, write the restricted domain.
G(t)=t+1t2β6tβ7
Reduced G(t)=
64.
Simplify the function formula, and if applicable, write the restricted domain.
h(x)=xβ5x2+xβ30
Reduced h(x)=
65.
Simplify the function formula, and if applicable, write the restricted domain.
K(x)=x3β81xx3+11x2+18x
Reduced K(x)=
66.
Simplify the function formula, and if applicable, write the restricted domain.
G(y)=y3β9yy3+13y2+30y
Reduced G(y)=
67.
Simplify the function formula, and if applicable, write the restricted domain.
h(y)=y4+4y3+4y23y4+5y3β2y2
Reduced h(y)=
68.
Simplify the function formula, and if applicable, write the restricted domain.
K(r)=r4β8r3+16r23r4β11r3β4r2
Reduced K(r)=
69.
Simplify the function formula, and if applicable, write the restricted domain.
G(r)=3r3+r23r3β11r2β4r
Reduced G(r)=
70.
Simplify the function formula, and if applicable, write the restricted domain.
g(r)=5r3+3r25r3β22r2β15r
Reduced g(r)=
Multiplying and Dividing Rational Expressions with One Variable
71.
Select all correct equations:
9β xy=9x9y
βxy=βxβy
9β xy=x9y
βxy=βxy
9β xy=9xy
βxy=xβy
72.
Select all correct equations:
10β xy=10x10y
βxy=βxβy
βxy=βxy
10β xy=10xy
10β xy=x10y
βxy=xβy
73.
Simplify the following expressions, and if applicable, write the restricted domain.
βx4x+4β x3=
βx4x+4β 1x3=
74.
Simplify the following expressions, and if applicable, write the restricted domain.
βy4y+4β y2=
βy4y+4β 1y2=
75.
Simplify this expression, and if applicable, write the restricted domain.
y2βyβ2y+4β 5y+20y+1=
76.
Simplify this expression, and if applicable, write the restricted domain.
y2+7y+12yβ6β 5yβ30y+4=
77.
Simplify this expression, and if applicable, write the restricted domain.
r2β9rr2β9β r2β3rr2β11r+18=
78.
Simplify this expression, and if applicable, write the restricted domain.
r2β9rr2β9β r2β3rr2β7rβ18=
79.
Simplify this expression, and if applicable, write the restricted domain.
12rβ12β20β25rβ5r2β r2+8r+164r2β4r=
80.
Simplify this expression, and if applicable, write the restricted domain.
6tβ2428β21tβ7t2β t2β2t+12t2β8t=
81.
Simplify this expression, and if applicable, write the restricted domain.
6t2β11t+520t3β50t2β 10t2β4t336t2β25=
82.
Simplify this expression, and if applicable, write the restricted domain.
5x2+(β1)xβ4126x2β105xβ 15xβ18x225x2β16=
83.
Simplify this expression, and if applicable, write the restricted domain.
xxβ6Γ·3x2=
84.
Simplify this expression, and if applicable, write the restricted domain.
yy+10Γ·5y2=
85.
Simplify this expression, and if applicable, write the restricted domain.
8yΓ·2y3=
86.
Simplify this expression, and if applicable, write the restricted domain.
12rΓ·3r2=
87.
Simplify this expression, and if applicable, write the restricted domain.
(2rβ6)Γ·(4rβ12)=
88.
Simplify this expression, and if applicable, write the restricted domain.
(4rβ12)Γ·(24rβ72)=
89.
Simplify this expression, and if applicable, write the restricted domain.
25t2β365t2+(β9)t+(β18)Γ·(6β5t)=
90.
Simplify this expression, and if applicable, write the restricted domain.
4t2β492t2+11t+14Γ·(7β2t)=
91.
Simplify this expression, and if applicable, write the restricted domain.
x4x2+6xΓ·1x2+xβ30=
92.
Simplify this expression, and if applicable, write the restricted domain.
x3x2β3xΓ·1x2+xβ12=
93.
Simplify this expression, and if applicable, write the restricted domain.
5a+1aa+1a =
94.
Simplify this expression, and if applicable, write the restricted domain.
10a+10aa+7a =
95.
Simplify this expression, and if applicable, write the restricted domain.
u(uβ6)25uu2β36=
96.
Simplify this expression, and if applicable, write the restricted domain.
r(rβ3)29rr2β9=
97.
Simplify this expression, and if applicable, write the restricted domain.
x2+3xx2β16Γ·x2β9x2+2xβ8=
98.
Simplify this expression, and if applicable, write the restricted domain.
x2+4xx2β1Γ·x2β16x2β4xβ5=
Multiplying and Dividing Rational Expressions with More Than One Variable
99.
Simplify this expression.
8(t+x)tβxβ tβx2(2t+x)=
100.
Simplify this expression.
12(x+t)xβtβ xβt4(2x+t)=
101.
Simplify this expression.
4x3y23x4β 9x4y28y5=
102.
Simplify this expression.
5yx3yβ 3y2x325x5=
103.
Simplify this expression.
y2+9yt+20t2y+tβ 2y+2ty+5t=
104.
Simplify this expression.
r2+ryβ12y2r+6yβ 3r+18yrβ3y=
105.
Simplify this expression.
rx104Γ·rx58=
106.
Simplify this expression.
r3t26Γ·r3t12=
107.
Simplify this expression.
(t4β4t3y+4t2y2)Γ·(t5β2t4y)=
108.
Simplify this expression.
(t3+8t2x+16tx2)Γ·(t5+4t4x)=
109.
Simplify this expression.
1x2β10xr+24r2Γ·x2x2β4xr=
110.
Simplify this expression.
1x2+5xy+6y2Γ·x5x2+2xy=
111.
Simplify this expression.
y5y2xβ6yΓ·1y2x2β7yx+6=
112.
Simplify this expression.
y3y2rβ4yΓ·1y2r2+2yrβ24=
113.
Simplify this expression.
36y4t2y+10tΓ·6y5ty2β100t2=
114.
Simplify this expression.
15r3t4r+9tΓ·3r8tr2β81t2=
115.
Simplify this expression.
pq5p4q2=
116.
Simplify this expression.
mn4m3n2=
117.
Simplify this expression.
mn210km6nk=
118.
Simplify this expression.
xy27zx10yz=
Challenge
119.
Simplify the following: 1x+1Γ·x+2x+1Γ·x+3x+2Γ·x+4x+3Γ·β―Γ·x+35x+34. For this exercise, you do not have to write the restricted domain of the simplified expression.