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Rational Expressions
Mathematics 17Institute of Mathematics
Lecture 4
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Outline
1 Simplifying Rational Expressions
2 Multiplication of Rational Expressions
3 Division of Rational Expressions
4 Addition/Subtraction of Rational Expressions
5 Complex Fractions
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Rational Expressions
DenitionA rational expression is a quotient of two polynomials.A rational expression is well-dened if its denominator is not equal to zero.
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Rational Expressions
DenitionA rational expression is a quotient of two polynomials.A rational expression is well-dened if its denominator is not equal to zero.
Example:
9x 2 z 3
x 2 + y ; x2 + y = 0
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Rational Expressions
DenitionA rational expression is a quotient of two polynomials.A rational expression is well-dened if its denominator is not equal to zero.
Example:
9x 2 z 3
x 2 + y ; x2 + y = 0
11a 2 c + b2
5 + c ; c = − 5
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Simpliying Rational Expressions
Recall: If a, b, c ∈R , b, c = 0 , then acbc
= ab
.
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Simpliying Rational Expressions
Recall: If a, b, c ∈R , b, c = 0 , then acbc
= ab
.
Lowest Terms
A rational expression is in lowest terms if the numerator and denominatorhave no common factor other than 1 and -1.
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Simpliying Rational Expressions
Recall: If a, b, c ∈R , b, c = 0 , then acbc
= ab
.
Lowest Terms
A rational expression is in lowest terms if the numerator and denominatorhave no common factor other than 1 and -1.
How to reduce a rational expression to lowest terms:
Factor both numerator and denominator.Cancel the common factors.
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
2
(x 2 − 16)(x 2 − 4x + 16)x 3 + 64
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 5 / 16
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
2
(x 2 − 16)(x 2 − 4x + 16)x 3 + 64
= (x − 4)(x + 4)
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
2
(x 2 − 16)(x 2 − 4x + 16)x 3 + 64
= (x − 4)(x + 4)( x 2 − 4x + 16)
(x + 4)
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
2
(x 2 − 16)(x 2 − 4x + 16)x 3 + 64
= (x − 4)(x + 4)( x 2 − 4x + 16)
(x + 4)( x 2 − 4x + 16)
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Examples:1
16x 3 y 8
24x 7 y 5 = 1624
· x3
x 7 · y8
y5 = 2y3
3x 4
2
(x 2 − 16)(x 2 − 4x + 16)x 3 + 64
= (x − 4)(x + 4)( x 2 − 4x + 16)
(x + 4)( x 2 − 4x + 16)= x − 4
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
l l f l
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
M l i li i f R i l E i
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
M l i li i f R i l E i
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 ·
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
M lti li ti f R ti l E i
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
M lti li ti f R ti l E i
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
5 ·
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
Multiplication of Rational Expressions
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
5 · (7)(4)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
Multiplication of Rational Expressions
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
5 · (7)(4) · x
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
Multiplication of Rational Expressions
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
5 · (7)(4) · x ·
1y 2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
Multiplication of Rational Expressions
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Multiplication of Rational Expressions
Recall:ab
· cd
= acbd
; b, d = 0
Example: 7x 4
5y 4 · 15y2
28x 3 = 7 · 15
5 · 28 ·
x4
x 3 · y2
y4
= 7 · (5)(3)
5 · (7)(4) · x ·
1y 2
= 3x4y2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 6 / 16
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
= (x − 4)(x + 3)
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
= (x − 4)(x + 3)
2(x − 1) ·
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
= (x − 4)(x + 3)
2(x − 1) ·
5(x − 1)
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
= (x − 4)(x + 3)
2(x − 1) ·
5(x − 1)(x − 3)(x + 3)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 7 / 16
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Example:
x 2 − x − 122x − 2
· 5x − 5x 2 − 9
= (x − 4)(x + 3)
2(x − 1) ·
5(x − 1)(x − 3)(x + 3)
= 5(x − 4)
2(x − 3)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 7 / 16
Division of Rational Expressions
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p
Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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p
Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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p
Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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p
Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
= (x − 3)(x − 4)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
= (x − 3)(x − 4)
x − 2 ·
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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Recall: ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
= (x − 3)(x − 4)
x − 2 ·
(x − 2)(x + 2)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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Recall:ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
= (x − 3)(x − 4)
x − 2 ·
(x − 2)(x + 2)(3 − x )(3 + x)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Division of Rational Expressions
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Recall:ab
÷ cd
= a
b ·
dc
; b,c,d = 0
Example:
x 2 − 7x + 12x − 2
÷ 9 − x 2
x 2 − 4 =
x2 − 7x + 12x − 2
· x 2 − 49 − x 2
= (x − 3)(x − 4)
x − 2 ·
(x − 2)(x + 2)(3 − x )(3 + x)
= − (x − 4)(x + 2)(x + 3)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 8 / 16
Least Common Denominator
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DenitionThe least common denominator (LCD) of two or more rational expressionsis the polynomial of smallest degree that is a multiple of all the givendenominators.
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Least Common Denominator
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DenitionThe least common denominator (LCD) of two or more rational expressionsis the polynomial of smallest degree that is a multiple of all the givendenominators.
Examples:
13x − 8
4x 2 and 2x + 1
x 3 :
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Least Common Denominator
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DenitionThe least common denominator (LCD) of two or more rational expressionsis the polynomial of smallest degree that is a multiple of all the givendenominators.
Examples:
13x − 8
4x 2 and 2x + 1
x 3 : LCD = 4x 3
2 x − 2x 2 − 1 and x + 2(x − 1)2 :
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 9 / 16
Least Common Denominator
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DenitionThe least common denominator (LCD) of two or more rational expressionsis the polynomial of smallest degree that is a multiple of all the givendenominators.
Examples:
13x − 8
4x 2 and 2x + 1
x 3 : LCD = 4x 3
2 x − 2x 2 − 1 and x + 2(x − 1)2 : LCD = ( x − 1)2 (x + 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 9 / 16
Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3
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Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3 =
(3x − 8)x4x 3
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Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3 =
(3x − 8)x +4x 3
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Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3 =
(3x − 8)x + 4(2 x + 1)4x 3
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 10 / 16
Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3 =
(3x − 8)x + 4(2 x + 1)4x 3
= 3x 2 − 8x + 8 x + 4
4x 3
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 10 / 16
Addition/Subtraction of Rational Expressions
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Example 1:
3x − 84x 2 +
2x + 1x 3 =
(3x − 8)x + 4(2 x + 1)4x 3
= 3x 2 − 8x + 8 x + 4
4x 3
= 3x 2 + 4
4x3
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
(x − 1)2 (x + 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 11 / 16
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
[(x − 1)(x − 2)](x − 1)2 (x + 1)
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
[(x − 1)(x − 2)] −(x − 1)2 (x + 1)
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
[(x − 1)(x − 2)] − [(x + 1)( x + 2)](x − 1)2 (x + 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 11 / 16
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
[(x − 1)(x − 2)] − [(x + 1)( x + 2)](x − 1)2 (x + 1)
= [x 2 − 3x + 2] − [x 2 + 3 x + 2]
(x − 1)2 (x + 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 11 / 16
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Example 2:
x − 2x 2 − 1
− x + 2(x − 1)2 =
[(x − 1)(x − 2)] − [(x + 1)( x + 2)](x − 1)2 (x + 1)
= [x 2 − 3x + 2] − [x 2 + 3 x + 2]
(x − 1)2 (x + 1)
= − 6x
(x − 1)2 (x + 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 11 / 16
Complex Fractions
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Complex fractions are fractions which contain one or more fractions in thenumerator or in the denominator or in both.
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Complex Fractions
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Complex fractions are fractions which contain one or more fractions in thenumerator or in the denominator or in both.
a − ab
c − cb
xy
− yx
xy
+ 2 + yx
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Simplifying Complex Fractions
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Example 1:
a − ab
c − cb
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Simplifying Complex Fractions
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Example 1:
a − ab
c − cb
=
ab − ab
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
Simplifying Complex Fractions
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Example 1:
a − ab
c − cb
=
ab − ab
cb − cb
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
Simplifying Complex Fractions
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Example 1:
a − ab
c − cb
=
ab − ab
cb − cb
= ab − a
b ·
bcb − c
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
Simplifying Complex Fractions
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Example 1:
a − abc −
cb
=
ab − ab
cb − cb
= ab − a
b ·
bcb − c
= ab − a
cb − c
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
Simplifying Complex Fractions
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Example 1:
a − abc −
cb
=
ab − ab
cb − cb
= ab − a
b ·
bcb − c
= ab − a
cb − c =
a(b − 1)c(b − 1)
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
Simplifying Complex Fractions
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Example 1:
a − abc −
cb
=
ab − ab
cb − cb
= ab − a
b ·
bcb − c
= ab − a
cb − c =
a(b − 1)c(b − 1)
= a
c
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 13 / 16
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Example 2: Multiplying and dividing by the LCD
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Example 2: Multiplying and dividing by the LCD
xy
− yx
xy
+ 2 + yx
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 14 / 16
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Example 2: Multiplying and dividing by the LCD
xy
− yx
xy
+ 2 + yx
=
xy
− yx
xy
xy
+ 2 + yx
xy
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 14 / 16
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Example 2: Multiplying and dividing by the LCD
xy
− yx
xy
+ 2 + yx
=
xy
− yx
xy
xy
+ 2 + yx
xy=
x2 − y2
x 2 + 2 xy + y2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 14 / 16
l l l d d d b h
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Example 2: Multiplying and dividing by the LCD
xy
− yx
xy
+ 2 + yx
=
xy
− yx
xy
xy
+ 2 + yx
xy=
x2 − y2
x 2 + 2 xy + y2
= (x + y)(x − y)
(x + y)2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 14 / 16
E l 2 M l i l i d di idi b h LCD
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Example 2: Multiplying and dividing by the LCD
xy
− yx
xy
+ 2 + yx
=
xy
− yx
xy
xy
+ 2 + yx
xy=
x2 − y2
x 2 + 2 xy + y2
= (x + y)(x − y)
(x + y)2 = x − yx + y
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 14 / 16
Exercises:
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Reduce the following rational expressions to lowest terms.
1a 2 − b2
a 3 − b32
x 3 − x 2 y + xy 2 − y 3
x 6 + y6
Perform the following operations and simplify.
1 x
x 2 − 1 − 3x + 1 ÷
2x 2 − x − 3x 3 − 1
2 x
x + y +
yx − y
·x 2 − xyx 4 − y4 ÷
xx 2 + 2 xy + y2
3m + 2 s
m 2 − 5ms + 6 s 2 +
m + 2 s
m 2 − 3ms + 2 s 2 −
m − 4s
m 2 − 4ms + 3 s 2
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 15 / 16
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Exercises:
Simplify the following:
1
3 p + q
+ 1
p − 2q
1 + p − q p − 2q
2
2 + 2c
2 − 22 +
4c − 4
Math 17 (Inst. of Mathematics) Rational Expressions Lec 4 16 / 16