Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Math 1010
Section 6.1: Rational Expressions and Functions
Definition of a rational expression
Let u and v be polynomials. The algebraic expression
u
v
is a rational expression. The domain of this rational expression is the set of all real numbers for which v 6= 0.
Definition of a rational function
Let u(x) and v(x) be polynomial functions. The function
f (x) =
u(x)
v(x)
is a rational function. The domain of f is the set of all real numbers for which v(x) 6= 0.
Ex.1
Find the domain of each rational function.
(1) f (x) = 4−x
x
(2) g(x) = 1−x
x+6
Ex.2
Find the domain of each rational function.
(1) f (x) = x25x
−16
(2) g(x) = x23x−1
−2x−3
1
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Math 1010
Definition of a rational expression
Let u, v, and w be real numbers, variables, or algebraic expressions such that v 6= 0 and w 6= 0. Then
uw
u
=
vw
v
Ex.3
Simplify the rational expression
2x3 − 6x
6x2
Ex.4
Simplify the rational expression
x2 + 2x − 15
3x − 9
2
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.5
Simplify the rational expression
x3 − 16x
x2 − 2x − 8
Ex.6
Simplify the rational expression
2x2 − 9x + 4
12 + x − x2
3
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.7
Simplify the rational expression
4x3 y − 5xy 2
2xy
Ex.8
Simplify the rational expression
4x2 y − y 3
2x2 y − xy 2
4
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Math 1010
Section 6.2: Multiplying and Dividing Rational Expressions
Multiplying rational expressions
Let u, v, w, and z be real numbers, variables, or algebraic expressions such that v 6= 0 and z 6= 0. Then
u w
uw
· =
v z
vz
Simplify the fraction if possible.
Ex.1
Multiply the rational expressions
4x3 y −6x2 y 2
·
3xy 4
10x4
Ex.2
Multiply the rational expressions
x
x−4
·
5x2 − 20x 2x2 + x − 3
5
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.3
Multiply the rational expressions
x2 + x − 6
4x2 − 4x
·
x2 + 2x − 3
4x
Ex.4
Multiply the rational expressions
x2 − 3x + 2
3x
2x + 4
·
·
x+2
x − 2 x2 − 5x
6
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Math 1010
Multiplying rational expressions
Let u, v, w, and z be real numbers, variables, or algebraic expressions such that v 6= 0, w 6= 0, and z 6= 0.
Then
u w
u z
uz
÷ = · =
v
z
v w
vw
Simplify the fraction if possible.
Ex.5
Divide the rational expressions
x
4
÷
x+3 x−1
Ex.6
Divide the rational expressions
x2 − 2x
2x
÷ 2
3x − 12 x − 6x + 8
7
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.7
Divide the rational expressions
2x2 − 3xy + y 2
x2 − y 2
÷
2x + 2y
6x + 2y
Section 6.3: Adding and Subtracting Rational Expressions
Multiplying rational expressions
Let u, v, w be real numbers, variables, or algebraic expressions such that w 6= 0. Then
v
u+v
u
+ =
w w
w
v
u−v
u
− =
w w
w
Ex.1
Combine and simplify the following rational expressions
x 5−x
+
4
4
8
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.2
Combine and simplify the following rational expressions
x
3
−
x2 − 2x − 3 x2 − 2x − 3
Ex.3
Combine and simplify the following rational expressions
x2 − 26 2x + 4 10 + x
−
+
x−5
x−5
x−5
9
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.4
Find the least common multiple of
(1) x2 − x and 2x − 2
(2) 3x2 + 6x and x2 + 4x + 4
(3) 5x − 25 and 2x2 − 9x − 5
Ex.5
Add the rational expressions
5
7
+
6x 8x
Ex.6
Subtract the rational expressions
3
5
−
x−3 x+2
10
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.7
Add the rational expressions
3
6x
+
x2 − 4 2 − x
Ex.8
Subtract the rational expressions
x
1
−
x2 − 5x + 6 x2 − x − 2
11
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.9
Combine and simplify the following rational expressions
4x
x
2
+
−
x2 − 16 x + 4 x
Section 6.4: Complex Fractions
Definition of complex fractions
A complex fraction is a fraction that has a fraction as numerator, or denominator, or both.
Ex.1
Simplify the complex fraction
4
( 18
)
2
(3)
12
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.2
Simplify the complex fraction
3
4y
( (5x)
2)
2
( (2y)
10x3 )
Ex.3
Simplify the complex fraction
( x+1
x+2 )
( x+1
x+5 )
13
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.4
Simplify the complex fraction
(x
2
+4x+3
x−2 )
2x + 6
Ex.5
Simplify the complex fraction
( x3 + 23 )
(1 − x2 )
14
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.6
Simplify the complex fraction
2
( x+2
)
3
( x+2
+ x2 )
Ex.7
Simplify the complex fraction
5 + x−2
8x−1 + x
15
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Section 6.5: Dividing Polynomials
Ex.1
Perform the division and simplify
12x2 − 20x + 8
4x
Ex.2
Divide x2 + 2x + 4 by x − 1.
16
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.3
Divide −13x3 + 10x4 + 8x − 7x2 + 4 by 3 − 2x.
Ex.4
Divide x3 − 2 by x − 1.
17
Math 1010
Chapter 6: Rational Expr., Eq., and Functions Lecture notes
Ex.5
Divide x4 + 6x3 + 6x2 − 10x − 3 by x2 + 2x − 3.
18
Math 1010