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Basic Properties
of Rational Expressions
A rational expression is any expression of the form
Q P
where P and Q are polynomials and Q
- In the following
properties, no denominator is allowed to be zero.
Basic A fraction is not defined when the denominator is zero!
Property:
Example:
Q P
S R
if and only if P • S = Q • R
Q P
Q • RP • R
Q P
Q–P
–Q P
–Q–P
Q P
Q–P
–Q P
–Q–P
Notice that
P –P
Since it is true that
x – a
It must also be true that
a – x x – a
W A R N I N G :
You must reduce only factors!!
If the terms are not factors,
they cannot be factored out.
Nonsense like "canceling out" nonfactors will not be tolerated!!
2x + 8 2
x + 4
x (^2)
x
x + 3
The Domain of a Rational Function:
Unless we are told otherwise, we assume that the domain of a
defined.function is the set of all real numbers for which the function is
A rational expression is undefined when the denominator is
zero.
Hence when you need to find the domain of a rational
Examples: denominator to be zero.function, you need to determine all values (if any) which cause the
Find the domain of the following functions:
a) f(x) = 2x + 4
b) g(x)
x – 4x + 5
c) h(x) =
x 3 2x + 5
d) F(x) =
x
x (^2)
Solution:
a) ℜ
(all real numbers)
b) {x
(^) ε (^) ℜ (^) | x (^) ≠ (^) 4 }
(all real numbers except 4)
c) {x
(^) ε (^) ℜ (^) | x (^) ≠ (^) 0, 2, –2 }
(all real numbers except 0, 2, –2)
d) (^) ℜ
(all real numbers)
Multiplication and Division of
Rational Expressions
To multiply two rational expression, multiply the numerators
together and multiply the denominators together.
Q P
S R
Q • S P • R
To divide two rational expressions, invert the one immediately
after the division sign, and do a rational expression multiplication.
Q P
÷
S R
Q P
R S
Q • RP • S
Example:
Perform the indicated operation and simplify: x (^2)
x (^2)
÷
x 2 4x + 2
2x (^2)
x (^2)
x (^2)
x (^2)
x (^2)
4x + 2
2x (^2)
x (^2)
(x – 5)(x + 2) (x + 3)(x – 1)
2(2x + 1) x(x – 1)
(2x + 1)(x – 5)
(x – 1)
(^2)
2(x + 2)x(x + 3)
R
ule for
A
dding or
S
ubtracting
F
ractions with
U
nequal
D
enominators
(FLEAS)
F
actor the rational expression.
- Find the
L
east Common Denominator (LCD).
E
qualize each denominators by replacing each fraction
with an equivalent one whose denominator is the LCD.
A
dd or
S
ubtract using RASFED.
Example:
Examples:
Perform the indicated operation and simplify.
a)
2
x 2
1
x 2
=
2
(x + 1)(x – 1)
1
(x + 1)
2
=
2(x + 1)
(x + 1)
(x – 1)^2
x – 1
(x + 1)
(x – 1)^2
= (x + 1)2(x + 1) – (x – 1)
(x – 1)^2
= (x + 1)2x + 2 – x + 1
(x – 1)^2
=
x + 3
(x + 1)
(x – 1)^2
b)
x – 2 x + 2
x 2
= x – 2 x + 2
x 2
(x + 2)(x – 2)
=
(x + 2)
2
(x + 2)(x – 2)
x 2
(x + 2)(x – 2)
= (x (^2)
(^2)
(x + 2)(x – 2)
= x 2
(^2)
(x + 2)(x – 2)
=
2x + 4
(x + 2)(x – 2)
=
2(x + 2)
(x + 2)(x – 2)
=
2
(x – 2)
Complex Fractions
A simple fraction is any rational expression whose numerator
and denominator contain no rational expression.
A complex fraction is any rational expression whose numerator
or denominator contains a rational expression.
To simplify complex fractions:
S t e p 1 :
Identify
all
fractions
in
the
numerator
and
denominator and find the LCD.
Step 2: Multiply the numerator and denominator by the LCD.
a) Examples:
^
^
b)
x + y
x
x 1 x + y
y^1
xy xy (x + y) ^
x 1
y^1
xy (x + y) y + x
xy
Example:
Simplify the following:
a)
y 2 1
y^1
y 2
y 2 1
y (^2)
y^1
9y (^2)
3y (^2)
(3y + 1)(3y – 1)
y(3y – 1)
3y + 1 y
b)
x + h
x^1
h
x(x + h)
^
x + h
x^1
x(x + h) h
x(x + h) hx – (x + h)
x(x + h) hx – x – h
–h
x(x + h) h
x(x + h)
Examples:
Calculate the indicated quotients by long division:
a)
x (^3)
(^2)
x + 2
b)
x (^4)
(^2)
x (^2)
c) 6x (^4)
2x – 1
Synthetic Division of Polynomials
When you divide a polynomial by a
(^) linear polynomial with
linear coefficient 1
, we can perform the division by using only the
necessary coefficients.
Step 1.
Write the opposite of the constant term of the divisor
by itself.
Write all of the coefficient of the dividend (using zero
when terms are missing, of course).
Step 2.
Bring down the first term of the dividend.
This also
becomes the current term.
Step 3.
b. Put the product under the a. Multiply the current term by the divisor term.
(^) next
(^) term of the dividend.
c. Add the result. The result becomes the current term.
Step 4.
If there is another term of the dividend, then go to
Step 3. Otherwise, go to Step 5.
Step 5.
The constants of the bottom line are the coefficients of
the quotient and the remainder.
Examples:
Calculate the indicated quotients by synthetic division:
a)
x (^3)
(^2)
x + 2
b)
x (^4)
(^2)
x – 3
c) x (^4)
x + 3
Remainder Theorem
When you divide a polynomial P(x) by the factor x – c, the
remainder is P(c).
Thus we sometimes evaluate a polynomial P(x)
Examples 1: when x = c by performing the appropriate synthetic division.
Let P(x) = 2x
(^3)
(^2)
b) Find the remainder when P(x) is divided by x – 2.a) By direct substitution, evaluate P(2).
Examples 2:
Let P(x) = 4x
(^6)
(^5)
(^4)
(^). Find P(4) (^2)
c)
y – 3 y – 2
y (^2)
y – 3 y – 2
(y + 3)(y – 3)
(y + 3)(y – 3)
y – 3 y – 2
= (y + 3)(y – 3)
^
(y + 3)(y – 3)
(y + 3)(y – 2) = (y + 3)(y – 3) – 2
y (^2)
(^2)
y = –
d)
x (^2)
= x 2 1
x (^2)
(x + 2)(x – 2)
x 2 1
x(x – 2)
x (^) (x + 2)(x – 2) (^2)
(x + 2)(x – 2)
= x
(^) (x + 2)(x – 2) (^2)
^
x 2 1 (^) +
x(x – 2)
2x (^2) = (x + 2)(x – 2)
2x (^2) = x (^2)
(^2)
2x (^2) = 2x
(^2)
x = 2–x = –
no solution
e)
x + 2
x + 3
(x + 2)(x + 3)
^
x + 2
= (x + 2)(x + 3)
x + 3
2(x + 2)(x + 3) + 10(x + 3) = 3(x + 2) 2(x (^2)
- 5x + 6) + 10x + 30 = 3x + 6
2x (^2)
- 10x + 12 + 10x + 30 = 3x + 6
2x (^2)
(2x + 9)(x + 4) = 0 x = –9/2, –
f)
y + 3 y + 1
y – 2y + 5
6y + 23
y (^2)
(y + 3)(y – 2)
y + 3 y + 1
y – 2y + 5
= (y + 3)(y – 2)
(y+3)(y–2)6y + 23
(y – 2)(y + 1) + (y + 3)(y + 5) = (y + 3)(y – 2) + (6y + 23)
y (^2)
(^2)
(^2)
2y (^2)
(^2)
y 2 = 4^
y = –2y = ±
Example: It takes Rosa, traveling at 50 mph, 45 minutes longer
Find the distance traveled.to go a certain distance than it takes Maria traveling at 60 mph.
distance
rate
time
Rosa
x
Maria
x
(^50) x
(^60) x
^
(^50) x
(^60) x
^
6x – 5x
x
225 miles
Example: Beth can travel 208 miles in the same length of time
it takes Anna to travel 192 miles.
If Beth’s speed is 4 mph greater
than Anna’s, find both rates.
distance
rate
time
Beth
x + 4
Anna
x
x + 4
x^192
x(x + 4)
^
x + 4 208
x(x + 4)
^
x + 4 208
208x
192(x + 4)
208x
192x + 768
16x
x
Anna 48 mphBeth 52 mph
Example:
John, Ralph, and Denny, working together, can
clean a store in 6 hours.
Working alone, Ralph takes twice as long
to clean the store as does John.
Denny needs three times as long as
does John. How long would it take each man working alone?
John
x hours
Ralph
2x hours
Denny
3x hours
together
6 hours
x 1
2x 1
3x 1
6x ^
x 1
2x 1
3x 1
6x ^ ^6 ^1
x
x
John
11 hours
Denny 33 hoursRalph 22 hours
Example: An inlet pipe on a swimming pool can be used to fill
the pool in 12 hours.
The drain pipe can be used to empty the pool
in 20 hours.
If the pool is empty and the drain pipe is accidentally
opened, how long will it take to fill the pool?
inlet pipe
12 hours
drain pipe
20 hours
together
x hours
x 1
60x
^
60x
^ x^1
5x – 3x
2x
x
30 hours
Example:
You can row, row, row your boat on a lake 5 miles
per hour.
On a river, it takes you the same time to row 5 miles
downstream as it does to row 3 miles upstream.
What is the speed
of the river current in miles per hour?
distance
rate
time
downstream
5 + x
upstream
5 – x
5 + x
5 – x
(5 + x)(5 – x)
^
5 + x
(5 + x)(5 – x)
^
5 – x
5(5 – x)
3(5 + x)
25 – 5x
15 + 3x
8x
x
mph
