Introduction to Rational Expressions

Show your work for each problem.

1.
Which of the following is a rational expression?
1. a)
2. b)
3. c)
4. d)
Answer:
is the only rational expression. The others contain non-polynomial elements: (radical), (exponential), (absolute value).
2.
For the expression , what value of makes it undefined?
Answer: 3
The expression is undefined when the denominator equals zero. gives .
3.
What is true about the expression ?
1. a)It is undefined when
2. b)It is a rational expression with no restrictions
3. c)It is not a rational expression
4. d)It is undefined when
Answer: It is a rational expression with no restrictions
Since the denominator is (a constant), it can never equal zero. This rational expression is defined for all real values of .
4.
Evaluate when .
Answer: 2
5.
Find the value that makes undefined.
Answer: -4
1. Which part do we set equal to zero - numerator or denominator? denominator
2. Write the equation: denominator = 0 x + 4 = 0
3. Solve for -4
6.
For the expression , how many values of make it undefined?
1. a)2
2. b)0
3. c)1
4. d)4
Answer: 2
when or . So there are 2 excluded values.
7.
For , one restriction is . What is the other restriction?
Answer: -5
. The factors are zero when or .
8.
Find all restrictions for .
Answer: 3, 4
1. Set the denominator equal to zero x^2 - 7x + 12 = 0
2. What two numbers multiply to 12 and add to -7? -3, -4
3. Factor the quadratic (x - 3)(x - 4)
4. What are the two restrictions? 3, 4
9.
Which statement about is correct?
1. a)It equals
2. b)It is undefined for all
3. c)It is undefined when
4. d)It equals 0 for all
Answer: It equals 0 for all
as long as the denominator isn't zero. The only restriction is .
10.
Evaluate when .
Answer: 7
Restriction is . Since , we can evaluate: .
11.
Find all restrictions for .
Answer: -3, 2
1. What two numbers multiply to -6 and add to 1? 3, -2
2. Factor the denominator (x + 3)(x - 2)
3. What values make each factor zero? -3, 2
12.
Which expression is NOT a rational expression?
1. a)
2. b)
3. c)
4. d)
Answer:
, which involves a negative exponent. is NOT a polynomial, so this is not a rational expression in standard form.
13.
How many restrictions does have?
Answer: 3
. Three factors give three restrictions: , , .
14.
Find all values that make undefined.
Answer: -2, 0, 2
1. Factor out the GCF from the denominator x(x^2 - 4)
2. Factor (difference of squares) (x + 2)(x - 2)
3. Write the fully factored denominator x(x + 2)(x - 2)
4. List all three restrictions -2, 0, 2
15.
For which value of is undefined but is defined?
1. a)
2. b)
3. c)
4. d)
Answer:
First expression: restrictions at . Second expression: restriction at . At , only the first is undefined while the second equals .
16.
For what values of is undefined?
Answer: -2, 2
1. Set the denominator equal to zero 2x^2 - 8 = 0
2. Factor out the GCF 2(x^2 - 4) = 0
3. Factor the difference of squares 2(x + 2)(x - 2) = 0
4. What are the two restrictions? -2, 2
17.
What is the result of for any real value of ?
1. a)
2. b)1
3. c)0
4. d)Undefined
Answer: 0
Since for all real , we have . The denominator is never zero, so the expression equals for all real .
18.
Evaluate when .
Answer: 10