1. Multiplying and Dividing Rational Expressions
Sign In
An error ocurred, try again later!
Chapter 4
1.
Multiplying and Dividing Rational Expressions
This lesson delves into the realm of rational expressions, specifically focusing on their multiplication and division. Rational expressions, essentially fractions where both the numerator and denominator are polynomials, have specific rules for operations. When multiplying, it's akin to multiplying fractions: numerators with numerators and denominators with denominators. Before performing these operations, it's beneficial to factor the expressions to simplify the process. Dividing rational expressions, on the other hand, involves multiplying by the reciprocal of the divisor. The lesson emphasizes the importance of understanding restrictions, as certain values can render a rational expression undefined. Through examples and detailed explanations, learners are equipped with the knowledge to handle these algebraic expressions effectively.
Show less Show more expand_more Lesson Settings & Tools
| | 12 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Multiplying and Dividing Rational Expressions
Slide of 12
As with rational numbers, it is possible to write two polynomial expressions as a ratio, so long as the divisor is not zero. This lesson will explore various aspects of this type of algebraic expression, including how mathematical operations such as multiplication and division can be applied to it.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Combined Fuel Efficiency
The distance that a vehicle can travel per one gallon of fuel is measured as its mile per gallon (mpg) fuel economy. Each car has two fuel economy numbers, one measuring its efficiency for city driving and the other for highway driving. The combined fuel economy C for x mpg in the city and y mpg on the highway is computed by the following formula.
C=21(x1+y1)1
a If a car travels 28 miles per gallon in the city and 36 mpg on the highway, what is the combined fuel economy for the car?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"mpg","answer":{"text":["31.5","\\dfrac{504}{16}","\\dfrac{63}{2}"]}}
b Let x be the city fuel economy for a new car model. If the highway fuel economy in terms of x is x−2010x, write a simplified expression for the combined fuel economy for such a car.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{20x}{x-10}"]}}
Discussion
Rational Expression
A rational expression is a fraction where both the numerator and the denominator are polynomials.
q(x)p(x)
x3+5xx2−7
A rational expression is said to be written in its simplest form if the numerator and denominator have no common factors. | Rational Expressions | |
|---|---|
| Not in Simplest Form | In Simplest Form |
| x(x−3)(y+2)xy | x+1x−1 |
| x2+1x4+x2 | x2−x−6x3+7 |
Notice that for some of the expressions shown in the table, there are some x-values that make the denominator 0. For example, the denominator of x+1x−1 is 0 when x=-1. Any value of a variable for which a rational expression is undefined is called an excluded value.
| Expression | Restriction | Excluded Value(s) |
|---|---|---|
| x+1x−1 | x+1=0 | x=-1 |
| x2−x−6x3+7 | x2−x−6=0 | x=-2 and x=3 |
| x(x−3)(y+2)xy | x(x−3)(y+2)=0 | x=0, x=3, and y=-2 |
| x2+1x4+x2 | There is no real number that makes x2+1 zero | None |
Simplifying a rational expression can remove some of the excluded values that appear in the original expression. A rational expression and its simplified form must have the same domain in order for them to be equivalent expressions. This means that the excluded values that are no longer visible in the simplified expression must still be declared.
| Equivalent Expressions | |
|---|---|
| Rational Expression | Simplified Form |
| (x+2)(x−3)x−3,x=-2,3 | x+21,x=-2,3 |
| x2−1x2+2x+1,x=-1,1 | x−1x+1,x=-1,1 |
| x2x3−2x2+x,x=0 | xx2−2x+1,x=0 |
Pop Quiz
Determining Values That Make a Rational Expression Undefined
A rational expression is undefined when its denominator is 0. The values that make the denominator of a rational expression equal to 0 are called excluded values because they are excluded from its domain. Determine the excluded values for the indicated rational expressions.
Discussion
Simplifying Rational Expressions
A rational expression is written in simplest form if the numerator and denominator have no factors in common. Rational expressions that are not in simplest form can be simplified by canceling out the common factors. There are many various methods of factoring polynomials to find any common factors. Consider an example expression.
9x−x3x2−6x+9
Rational expressions can be simplified in three steps.
1
Factor the Numerator and the Denominator
expand_more Check first if either the numerator or the denominator — or both — can be factored by the greatest common factor. In this example, the factor x is in both terms of the denominator.
Determine if it is possible to factor either the numerator or the denominator by using a difference of squares. Look for an expression in the form a2−b2, which can be factored as (a+b)(a−b). In the example, (9−x)2 can be factored using this rule.
See if either the numerator or the denominator is a perfect square trinomial. Look for an expression in the form a2±2ab+b2, which can be factored as (a±b)2. Here, the numerator is a perfect square trinomial.
In some cases, a negative sign can be factored out for one expression to have the same form as another factor. Here, the factor 3−x in the denominator is almost identical to the factor x−3 in the numerator.
Now both the numerator and denominator are completely factored.
9x−x3x2−6x+9
SplitIntoFactors
Split into factors
x⋅9−x⋅x2x2−6x+9
FactorOut
Factor out x
x(9−x2)x2−6x+9
x(9−x2)x2−6x+9
WritePow
Write as a power
x(32−x2)x2−6x+9
FacDiffSquares
a2−b2=(a+b)(a−b)
x(3+x)(3−x)x2−6x+9
x(3+x)(3−x)x2−6x+9
SplitIntoFactors
Split into factors
x(3+x)(3−x)x2−2x(3)+9
WritePow
Write as a power
x(3+x)(3−x)x2−2⋅x⋅3+32
FacNegPerfectSquare
a2−2ab+b2=(a−b)2
x(3+x)(3−x)(x−3)2
x(3+x)(3−x)(x−3)2
CommutativePropMult
Commutative Property of Multiplication
(3−x)x(3+x)(x−3)2
FactorOutNegSignSwap
a−b=-(b−a)
-(x−3)x(3+x)(x−3)2
2
List Restricted Values
expand_more Before simplifying the common factors, check if there are any restrictions on x.
-(x−3)x(3+x)(x−3)2
Note that the expression is undefined when x=-3, x=0, or x=3. 3
Cancel Out Common Factors
expand_more In this expression, (x−3) is the common factor of the numerator and the denominator and can be eliminated.
The rational expression is now written in its simplest form. In order for the given expression and the simplified expression to be equivalent, the domain of the simplified expression should be restricted by excluding x=3.
-x(3+x)x−3, x=3
Since the restriction x=3 cannot be seen from the simplified expression, it should be written. The other restrictions are evident from the simplified expression. Example
Finding the Domain of Rational Expressions
Kevin and Zosia are asked to find the values that make the following rational expression undefined.
Who is correct?
x2+2x−35x2+12x+35
They disagree about the domain of the rational expression. {"type":"choice","form":{"alts":["Kevin","Zosia"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
A rational expression is undefined for values that make its denominator zero. Therefore, those values should be excluded from the domain.
Solution
A rational expression is undefined when the denominator equals 0.
Now the Zero Product Property will be applied to find all excluded values.
The values x=-7 and x=5 make the denominator equal to 0. As a result, they make the rational expression undefined. Therefore, Zosia is correct in saying that the domain is all real numbers except -7 and 5. Kevin may have simplified the expression before identifying the excluded values. This is a common mistake!
Note that x=5 is the value that makes the denominator of the simplified expression equal to 0. However, Kevin should have noted that -7 makes the original expression undefined. Since the original and simplified expressions are equivalent, both should have the same domain.
x2+2x−35x2+12x+35
To find the domain of the rational expression, the expression in the denominator is set equal to 0.
x2+2x−35=0
To solve the resulting equation for x, start by factoring the expression on the left-hand side.
x2+2x−35=0
(x+7)(x−5)=0
(x+7)(x−5)=0
▼
Solve using the Zero Product Property
ZeroProdProp
Use the Zero Product Property
x+7=0x−5=0(I)(II)
SubEqn
(I): LHS−7=RHS−7
x=-7x−5=0
AddEqn
(II): LHS+5=RHS+5
x=-7x=5
(x+7)(x−5)x2+12x+35
(x+7)(x−5)(x+7)(x+5)
▼
Simplify
(x−5)(x+5)
Discussion
Multiplying and Dividing Rational Expressions
Operations with rational numbers and rational expressions are similar.
Multiplying Rational Expressions
Multiplying rational expressions works the same way as multiplying fractions. The numerators and denominators are multiplied separately.
Q(x)P(x)⋅G(x)H(x)=Q(x)⋅G(x)P(x)⋅H(x)
7x+42x2−2x−15⋅x2−11x+30x2+9x+18
Three steps can be followed to multiply the rational expressions.
1
Factor Each Numerator and Denominator
expand_more Before multiplying, it is helpful to factor the numerators and denominators, if possible. Start with the first rational expression.
Now the numerator and denominator of the second expression will be factored.
x2−11+30x2+9x+18
x2−11x+30(x+3)(x+6)
(x−5)(x−6)(x+3)(x+6)
2
Multiply the Numerators and Multiply the Denominators
expand_more Now that both expressions have been factored, the numerators and denominators can be multiplied.
The product is undefined when x=-6, x=5, and when x=6.
7(x+6)(x+3)(x−5)⋅(x−5)(x−6)(x+3)(x+6)
MultFrac
Multiply fractions
7(x+6)(x−5)(x−6)(x+3)(x−5)(x+3)(x+6)
4
Cancel Out Common Factors
expand_more Note that (x−5) and (x+6) are common factors of the numerator and denominator. Finally, the product is simplified by canceling out common factors.
Considering the denominator of the simplified expression and any other denominator used, the values x=-6, x=5, and x=6 must be excluded from the domain of the simplified expression. In this expression it is clear that x=6 makes the denominator 0. Therefore, the fact that x=6 will not be expressed.
7(x+6)(x−5)(x−6)(x+3)(x−5)(x+3)(x+6)
CancelCommonFac
Cancel out common factors
7(x+6)(x−5)(x−6)(x+3)(x−5)(x+3)(x+6)
SimpQuot
Simplify quotient
7(x−6)(x+3)(x+3)
ProdToPowTwoFac
a⋅a=a2
7(x−6)(x+3)2
7(x−6)(x+3)2, x=-6, x=5
Dividing Rational Expressions
Dividing two rational expressions is the same as multiplying the first expression by the reciprocal of the second expression.
x2+3x5−x÷x2+10x+21x2−25
The process of dividing rationals expression can be completed in four steps.
1
Rewrite the Division as the Product of the Dividend and the Reciprocal of the Divisor
expand_more To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second.
x2+3x5−x÷x2+10x+21x2−25⇕x2+3x5−x⋅x2−25x2+10x+21
Once the quotient is expressed as a product, the remaining steps are the same as those for multiplying rational expressions. 2
Factor Each Numerator and Denominator
expand_more Factor the numerators and the denominators, if possible. The first expression will be factored first.
Now the numerator and the denominator of the second expression will be factored.
x2−25x2+10x+21
x2−25(x+3)(x+7)
▼
Factor the denominator
(x+5)(x−5)(x+3)(x+7)
3
Multiply the Numerators and Multiply the Denominators
expand_more Next, the numerators and denominators can be multiplied.
x(x+3)5−x⋅(x+5)(x−5)(x+3)(x+7)⇕x(x+3)(x+5)(x−5)(5−x)(x+3)(x+7)
This product is undefined when x=-5, x=-3, x=0, and x=5. Also, the values that make the divisor's denominator in the original quotient expression equal 0 should be excluded. These values are x=-7 and x=-3.
Excluded Values-7, -5, -3, 0, 5
4
Cancel Out Common Factors
expand_more Finally, the product is simplified by canceling out common factors.
Considering the denominator of the simplified expression and any other denominator used in this process, the values x=-7, x=-5, x=-3, x=0, and x=5 must be excluded from the domain of the expression. The simplified expression shows that x=0 and x=-5 must be excluded, so these values are not mentioned.
x(x+3)(x+5)(x−5)(5−x)(x+3)(x+7)
FactorOut
Factor out -1
x(x+3)(x+5)(x−5)-1(x−5)(x+3)(x+7)
CancelCommonFac
Cancel out common factors
x(x+3)(x+5)(x−5)-1(x−5)(x+3)(x+7)
SimpQuot
Simplify quotient
x(x+5)-1(x+7)
Distr
Distribute -1
x(x+5)-x−7
x(x+5)-x−7, x=-7, x=-3, x=5
Example
Representing Areas with Rational Expressions
Ramsha drew the plan of her house and labeled the sides, measured in meters, as shown.
a Write a simplified rational expression for the area of the house.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord Roboto-Regular\">m<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["\\dfrac{18x}{(x+9)(x+3)}","\\dfrac{18x}{x^2+12x+27}"]}}
b State any restrictions on x.
{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"ordermatters":false,"numinput":3,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-9","-3","3"]}}
Hint
a Start by factoring the numerator and the denominator of each rational expression.
b The denominator of a rational expression cannot be zero.
Solution
a Start by considering the formula for the area of a rectangle.
A=ℓw
According to the diagram, the length ℓ of the house is represented by x2+18x+812x2−6x and the width w by x2−99x+81. 
A=ℓw
SubstituteII
ℓ=x2+18x+812x2−6x, w=x2−99x+81
A=x2+18x+812x2−6x⋅x2−99x+81
▼
Factor the first expression
FactorOut
Factor out 2x
A=x2+18x+812x(x−3)⋅x2−99x+81
SplitIntoFactors
Split into factors
A=x2+2(9)x+812x(x−3)⋅x2−99x+81
CommutativePropMult
Commutative Property of Multiplication
A=x2+2x(9)+812x(x−3)⋅x2−99x+81
WritePow
Write as a power
A=x2+2x(9)+922x(x−3)⋅x2−99x+81
FacPosPerfectSquare
a2+2ab+b2=(a+b)2
A=(x+9)22x(x−3)⋅x2−99x+81
▼
Factor second expression
FactorOut
Factor out 9
A=(x+9)22x(x−3)⋅x2−99(x+9)
WritePow
Write as a power
A=(x+9)22x(x−3)⋅x2−329(x+9)
FacDiffSquares
a2−b2=(a+b)(a−b)
A=(x+9)22x(x−3)⋅(x+3)(x−3)9(x+9)
(x+9)22x(x−3)⋅(x+3)(x−3)9(x+9)
▼
Simplify
MultFrac
Multiply fractions
(x+9)2⋅(x+3)(x−3)2x(x−3)⋅9(x+9)
PowToProdTwoFac
a2=a⋅a
(x+9)(x+9)⋅(x+3)(x−3)2x(x−3)⋅9(x+9)
CancelCommonFac
Cancel out common factors
(x+9)(x+9)⋅(x+3)(x−3)2x(x−3)⋅9(x+9)
SimpQuot
Simplify quotient
(x+9)⋅(x+3)2x⋅9
Multiply
Multiply
(x+9)(x+3)18x
b To identify the restrictions, the denominator of the simplified expression and any other denominator used in this process will be analyzed. For simplicity, their factored forms will be used.
| Denominator | Restrictions on the Denominator | Restrictions on the Variable |
|---|---|---|
| (x+9)2 | (x+9)2=0 | x=-9 |
| (x+3)(x−3) | x+3=0 and x−3=0 | x=-3 and x=3 |
| (x+9)(x+3) | x+9=0 and x+3=0 | x=-9 and x=-3 |
x=-9, x=-3, x=3
Example
Determining the Type With the Best Efficiency Ratio
Companies aim to produce packaging using the lowest possible amount of material. They produce their packages in such a way that the ratio of the surface area S of a package to its volume V is as small as possible.
Efficiency Ratio:VS
A company is designing two different types of packages. The table shows the expressions for their surface areas and volumes. | Surface Area, S | Volume, V | |
|---|---|---|
| Type I | 4x4+12x3x3−x2 | x3+3x2x2−2x+1 |
| Type II | 4x2−28x+242x2−9x−18 | 2x2+x−3 |
a Find the efficiency ratio of each type.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<b>Type I:<\/b>","formTextAfter":null,"answer":{"text":["\\dfrac{x}{4(x-1)}","\\dfrac{x}{4x-4}"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<b>Type II:<\/b>","formTextAfter":null,"answer":{"text":["\\dfrac{1}{4(x-1)^2}","\\dfrac{1}{(2x-2)^2}","\\dfrac{1}{4x^2-8x+4}"]}}
b Which type is more efficient when x=2?
{"type":"choice","form":{"alts":["Type I","Type II","Both are equally efficient."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
a Start by factoring each denominator and numerator, if possible. Recall that all denominators must be different than 0.
b Substitute the given value for x into each rational expression found in Part A. The smaller the ratio, the more efficient the packaging.
Solution
a To find the efficiency ratios, the ratio of the surface area to the volume will be found for each packaging type. First, the efficiency ratio of Type I will be calculated.
4x4+12x3x3−x2÷x3+3x2x2−2x+1
To divide rational expressions, multiply the first expression by the reciprocal of the second expression.
4x4+12x3x3−x2÷x3+3x2x2−2x+1⇕4x4+12x3x3−x2⋅x2−2x+1x3+3x2
To multiply these expressions, begin by factoring the numerators and the denominators of each expression, if possible. Start with the first rational expression.
Now the second rational expression will be factored.
x2−2x+1x3+3x2
FactorOut
Factor out x2
x2−2x+1x2(x+3)
▼
Factor denominator
IdPropMult
Identity Property of Multiplication
x2−2x(1)+1x2(x+3)
WritePow
Write as a power
x2−2x(1)+12x2(x+3)
ExpandNegPerfectSquare
(a−b)2=a2−2ab+b2
(x−1)2x2(x+3)
4x3(x+3)x2(x−1)⋅(x−1)2x2(x+3)
MultFrac
Multiply fractions
4x3(x+3)(x−1)2x2(x−1)x2(x+3)
▼
Simplify
Write power as a product
4x2⋅x(x+3)(x−1)(x−1)x2(x−1)x⋅x(x+3)
CancelCommonFac
Cancel out common factors
4x2⋅x(x+3)(x−1)(x−1)x2(x−1)x⋅x(x+3)
SimpQuot
Simplify quotient
4(x−1)x
4x2−28x+242x2−9x−18÷(2x2+x−3)
To divide a rational expression by polynomial, the first expression is multiplied by the reciprocal of the polynomial. 4x2−28x+242x2−9x−18÷(2x2+x−3)⇕4x2−28x+242x2−9x−18⋅2x2+x−31
The table shows the steps taken to perform the multiplication. | Multiplying Rational Expressions | |
|---|---|
| Product | 4x2−28x+242x2−9x−18⋅2x2+x−31 |
| Factor | 4(x−1)(x−6)(2x+3)(x−6)⋅(2x+3)(x−1)1 |
| Multiply | 4(x−1)(x−6)(2x+3)(x−1)(2x+3)(x−6) |
| Cancel Out Common Factors | 4(x−1)(x−6)(2x+3)(x−1)(2x+3)(x−6) |
| Simplify | 4(x−1)21 |
The efficiency ratio of Type II is 4(x−1)21
b In Part A, the efficiency ratio of each type was found. Now, these ratios will be evaluated for x=2.
| Type I | Type II | |
|---|---|---|
| Efficiency Ratio | 4(x−1)x | 4(x−1)21 |
| Substitute | 4(2−1)2 | 4(2−1)21 |
| Evaluate | 21 | 41 |
Recall that the smaller the ratio, the more efficient the packaging. Therefore, Type II is more efficient because 41<21.
Discussion
Complex Fractions
A complex fraction is a rational expression where the numerator, denominator, or both, contain a rational expression.
m(x)q(x)r(x)p(x)
m(x)q(x) r(x)p(x) ⇔r(x)p(x)÷m(x)q(x)
As an example of a complex fraction, consider the following division of rational expressions. x+yx−y x2−y2x2−2x+4 ⇔ x2−y2x2−2x+4÷x+yx−y
Example
Surface Area to Volume Ratio of Penguins
Animals adapt to their environment. As a result of adaptations, the surface area and volume of animals vary depending on where they live. For example, penguins have a lower surface area to volume ratio to conserve their body heat.
External credits: @macrovector
SV= 3r−62πr3+2πr2h= 3r2−12r+12πr4h−2πr3h
Simplify the complex fraction VS. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["r","h"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{2(r+h)}{rh}","\\dfrac{2r+2h}{rh}"]}}
Hint
Start by rewriting the complex fraction as a division expression and then divide the rational expressions.
Solution
The ratio of the surface area S of a penguin to its volume V needs to be simplified.
Next, the numerators and denominators are multiplied and the common factors are eliminated.
3r2−12r+12πr4h−2πr3h 3r−62πr3+2πr2h
This complex fraction can be rewritten as a division expression.
3r−62πr3+2πr2h÷3r2−12r+12πr4h−2πr3h
Dividing two rational expressions is the same as multiplying the first expression by the reciprocal of the second expression.
3r−62πr3+2πr2h÷3r2−12r+12πr4h−2πr3h⇕3r−62πr3+2πr2h⋅πr4h−2πr3h3r2−12r+12
Now the multiplication can be performed. To do so, begin by factoring the numerators and denominators of each expression, if possible. The first expression will be factored first.
Now the second rational expression will be factored.
πr4h−2πr3h3r2−12r+12
▼
Factor the numerator
FactorOut
Factor out 3
πr4h−2πr3h3(r2−4r+4)
SplitIntoFactors
Split into factors
πr4h−2πr3h3(r2−2(2)r+4)
CommutativePropMult
Commutative Property of Multiplication
πr4h−2πr3h3(r2−2r(2)+4)
WritePow
Write as a power
πr4h−2πr3h3(r2−2r(2)+22)
ExpandNegPerfectSquare
(a−b)2=a2−2ab+b2
πr4h−2πr3h3(r−2)2
FactorOut
Factor out πr2h
πr3h(r−2)3(r−2)2
3(r−2)2πr2(r+h)⋅πr3h(r−2)3(r−2)2
MultFrac
Multiply fractions
3(r−2)πr3h(r−2)2πr2(r+h)3(r−2)2
▼
Simplify
Write power as a product
3(r−2)πr2rh(r−2)2πr2(r+h)3(r−2)(r−2)
CancelCommonFac
Cancel out common factors
3(r−2)πr2rh(r−2)2πr2(r+h)3(r−2)(r−2)
SimpQuot
Simplify quotient
rh2(r+h)
Closure
Finding Combined Fuel Efficiency of a New Car Model
With the methods seen in the lesson, the challenge given at the beginning can finally be solved. Recall the combined fuel economy formula.
C=21(x1+y1)1
In the formula, x represents miles per gallon (mpg) in the city and y represents miles per gallon on the highway. a If a car travels 28 mpg in the city and 36 mpg on the highway, what is the combined fuel economy for the car?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"mpg","answer":{"text":["31.5","\\dfrac{504}{16}","\\dfrac{63}{2}"]}}
b Let x be the city fuel economy for a new car model. If the highway fuel economy, in terms of x, is x−2010x, write a simplified expression for the combined fuel economy for such a car.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">C<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{20x}{x-10}"]}}
Hint
a Substitute the given values and evaluate the formula.
b Substitute the given expression for y and simplify it. When 1 is divided by a rational expression, the result is the reciprocal of the the rational expression.
Solution
a A car travels at 28 mpg in the city and 36 mpg on the highway. The combined fuel economy can be found by substituting these values into the given formula.
C=21(x1+y1)1
SubstituteII
x=28, y=36
C=21(281+361)1
▼
Simplify denominator
ExpandFrac
ba=b⋅9a⋅9
C=21(2529+361)1
ExpandFrac
ba=b⋅7a⋅7
C=21(2529+2527)1
AddFrac
Add fractions
C=21(25216)1
MultFrac
Multiply fractions
C=504161
DivOneByFracD
a/b1=ab
C=16504
CalcQuot
Calculate quotient
C=31.5
b In this case, the rational expression x−2010x will be substituted for y.
C=21(x1+x−2010x1)1
Recall that when 1 is divided by a rational expression, the result is the reciprocal of the the rational expression. This fact can be used twice to simplify the right-hand side of the equation.
C=21(x1+x−2010x1)1
▼
Simplify denominator
DivOneByFracSL
1/ba=ab
C=21(x1+10xx−20)1
ExpandFrac
ba=b⋅10a⋅10
C=21(10x10+10xx−20)1
AddFrac
Add fractions
C=21(10xx−10)1
MultFrac
Multiply fractions
C=20xx−101
DivOneByFracSL
1/ba=ab
C=x−1020x
Multiplying and Dividing Rational Expressions
Exercises
Simplify the rational expression.
3x2−4x6x3
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{6x^2}{3x-4}"]}}
3x39x3−x2
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{9x-1}{3x}"]}}
security
report
code
security
report
code
We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors. Let's do it!
We simplified the given expression.
Let's factor the numerator and denominator as much as possible. If they have any common factors, we will cancel them out.
We have simplified the expression.
security
report
code
Find the product of the rational expressions.
x36x2y3⋅10x3y23x2y
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{9y^2}{5x^2}"]}}
x66xy3⋅36xy4x4
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{1}{6x^2y}"]}}
security
report
code
security
report
code
We want to multiply the rational expressions. 6x^2y^3/x^3 * 3x^2y/10x^3y^2 We multiply numerator by numerator and denominator by denominator. Then, we can cancel out any common factors.
In a similar fashion, we will multiply the given rational expressions. 6xy^3/x^6 * x^4/36xy^4 Let's do it!
security
report
code
Find the quotient of the rational expressions.
y−x7x+7y÷5x−5y28
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["-\\dfrac{5(x+y)}{4}","\\dfrac{-5(x+y)}{4}","\\dfrac{-5x-5y}{4}","-\\dfrac{5x+5y}{4}","\\dfrac{5x+5y}{-4}"]}}
2x−105x÷x2+x−30x2+6x
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{5}{2}"]}}
security
report
code
security
report
code
We want to divide the rational expressions. 7x+7y/y-x÷ 28/5x-5y We will begin by rewriting the division as a product. To be more specific, we will multiply 7x+7yy-x by the reciprocal of 285x-5y. 7x+7y/y-x * 5x-5y/28 The remaining steps are the same as those for multiplying rational expressions. We can factor the numerators and denominators of each expression and cancel out any common factors. Let's start with the first expression.
Now, let's factor the second expression.
Finally, we can multiply the expressions and cancel out any common factors.
Even if the order of the steps we follow in dividing rational expressions changes, the result will not change. 5x/2x-10÷ x^2+6x/x^2+x-30 We can begin by factoring the numerators and denominators of each expression. Since the numerator of the first expression is already factored, we will factor only the denominator.
Now, let's factor the second expression.
We can now multiply the first expression by the reciprocal of the second expression. 5x/2(x-5) ÷ x(x+6)/(x-5)(x+6) ⇕ 5x/2(x-5) * (x-5)(x+6)/x(x+6) Finally, we cancel out any common factors.
security
report
code
3x3−15x2−5x+2518x4−50
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{2(3x^2+5)}{x-5}","\\dfrac{6x^2+10}{x-5}"]}}
{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"ordermatters":false,"numinput":3,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["5","\\sqrt{\\dfrac{5}{3}}","-\\sqrt{\\dfrac{5}{3}}"]}}
security
report
code
security
report
code
We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors.
The expression cannot be factored any further.
We will identify the restrictions on the variable from the denominator of the simplified expression and from any other denominator used. For simplicity, we will use their factored forms.
| Denominator | Restrictions on the Denominator | Restrictions on the Variable |
|---|---|---|
| (x-5)(3x^2-5) | x-5 ≠ 0 and 3x^2-5 ≠ 0 | x≠ 5 and x = ± sqrt(5/3) |
| x-5 | x-5≠ 0 | x≠ 5 |
We see that there are three unique restrictions on the variable x. x ≠ - sqrt(5/3), x ≠ sqrt(5/3), x ≠ 5
security
report
code
Which of the expressions does not belong with the other three?
A. B. C. D. x2+2x−3x2−xx2−1x−1⋅xx2+4x+3x2−9x2+2x−15÷xx+53xy−3xx+33y−31
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">A<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">B<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
security
report
code
security
report
code
A simplified rational expression will not have any common factors in the numerator and denominator. Let's try to look for any common factors in the given rational expressions one at a time.
Option A
Let's factor the numerator and denominator.
Option B
In option B, we are given a multiplication of two rational expressions. Let's factor the numerators and denominators of each expression before multiplying the rational expressions.
Now we can multiply and simplify common factors.
Option C
We need to find the quotient of the rational expressions. Since the divisor is fully factored, we only need to factor the dividend.
Recall that dividing by a rational expression is the same as multiplying by the reciprocal of the rational expression. The reciprocal of x+5x is xx+5.
Option D
Finally, we will simplify the given complex fraction.
Conclusion
After simplifying and performing operations, we see that only option B is simplified to a different expression. Therefore, the answer is option B.
| Given | Simplified Form | |
|---|---|---|
| Option A | x^2-x/x^2+2x-3 | x/x+3 |
| Option B | x-1/x^2-1 * x^2+4x+3/x | x+3/x |
| Option C | x^2+2x-15/x^2-9 ÷ x+5/x | x/x+3 |
| Option D | 13y-3/x+33xy-3x | x/x+3 |
security
report
code
Multiplying and Dividing Rational Expressions
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Loading content