Math 150 – Fall 2015
Chapter 1D
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Chapter 1D – Rational Expressions
Definition. A rational expression is the quotient pq of two polynomials, p and q.
Polynomials are also rational expressions with denominator 1.
Example 1. The following are rational expressions:
5x + 3
,
x2 − 7
3x3 − 4x + 10
,
x6 − 5x4 + 3x
x2
2
,
+3
x2 + 3x − 4
Example 2. These are not rational expressions:
√
3x2 − 7
sin x2
,
x2 − 4
3x5 − 2x2 + 3
Domain: The domain of a rational expression is all the values of the variable where
the expression is defined, which is all the values where the denominator is not zero.
Example 3. Find the domain of the following rational expressions:
1.
x2 −7x+8
3x−11
2.
3x
4x2 +36
Simplifying Rational Expressions: To simplify a rational expression, factor the
polynomials in the numerator and denominator and cancel the common factors. Make
sure to restrict the domain of the expression as necessary
Example 4. Simplify the rational expressions, and give the restrictions on x.
(a)
6x4 −9x3
3x
(b)
5x5 −40x2
x4 −4x2
(c)
3x4 +7x3 −20x2
5x4 +5x3 −60x2
Math 150 – Fall 2015
Chapter 1D
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Operations with Rational Expressions
Note. Rational expressions behave like fractions (just the numerator and denominator
are polynomials instead of numbers).
1. Multiplication of Rational Expressions Rational expressions multiply just like
fractions: multiply the numerators and multiply the denominators.
ac
a c
· =
b d
bd
Then simplify the rational expression.
Example 5. Perform the operation and simplify. List all restrictions on x.
(a)
2x−6
x−1
(b)
2x2 +x−6
x2 +4x−5
·
2x+4
x−3
·
x3 −3x2 +2x
4x2 −6x
·
x+5
x3 −8
2. Division of Rational Expressions Rationals expressions divide like fractions:
dividing by a rational expression is the same as multiplying by its reciprocal (the reciprocal switches the numerator and denominator).
c
a d
a
÷ = ·
b
d
b c
Example 6. Perform the operation and simplify. List all restrictions on x.
(a)
x2 −10x+21
x2 +6x+5
÷ (x2 + 2x − 15)
Math 150 – Fall 2015
(b)
x3 −27
2x4 −5x3
÷
x2 +x−12
x2 +5
Chapter 1D
·
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x4 +4x3
x2 −9
3. Addition and Subtraction of Rational Expressions Rational expressions add
and subtract just like fractions: you need to find a least common denominator (LCD)
and convert each fraction to a fraction whose denominator is the LCD.
To find the LCD of several fractions:
1. Factor all the denominators
2. The LCD is the product of all the prime factors of the denominators, and the
exponent of each prime factor is the highest exponent out of all the denominators.
Example 7. Perform the operation and simplify. List any restrictions on the variable.
(a)
2x+4
x−3
(b)
y−4
y 2 −2y
+
x−5
x+1
−
y 2 +3y+7
y 2 −3y+2
Math 150 – Fall 2015
Chapter 1D
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Compound Fractions: A compound fractions (or complex fraction) is an expression which contains nested fractions, usually one “main” fraction which will have
one or more fractions in its numerator and/or denominator.
To simplify a compound fraction:
1. Simplify the numerator and denominator of the main fraction.
2. Perform the division by multiplying by the reciprocal of the denominator.
Example 8. Simplify the expression and list any restrictions on the variables.
(a)
(b)
6
5
y − 2y+1
6
y +4
5(x+h)
−3
h
−5x−3