Simplifying Rational Expressions

Show your work for each problem.

1.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
, where .
2.
Simplify and enter the coefficient only (the number in front of ).
Answer: 3
. The coefficient is 3.
3.
Which expression is already in simplest form?
1. a)
2. b)
3. c)
4. d)
Answer:
In , the in the numerator is a term, not a factor, so it cannot be canceled. The other expressions can all be simplified further.
4.
Simplify
Answer: 2x^2y
1. What is ? 2
2. What is ? x^2
3. What is ? y
4. Combine: what is the simplified form? 2x^2y
5.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
. So , where .
6.
What is the restriction for after simplifying? (Enter the value that cannot equal)
Answer: -5
The original denominator is , which equals zero when . Even after simplifying to , we must exclude .
7.
Simplify
Answer: x + 2
1. Factor the numerator. Find two numbers that multiply to 2 and add to 3. (x + 1)(x + 2)
2. Identify the common factor between numerator and denominator. x + 1
3. After canceling the common factor, what is the simplified form? x + 2
8.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
, where and .
9.
Simplify
Answer: (x - 2)/(x + 2)
1. Factor the numerator (it's a perfect square trinomial). (x - 2)^2
2. Factor the denominator (difference of squares). (x + 2)(x - 2)
3. What is the common factor? x - 2
4. What is the simplified form? (x - 2)/(x + 2)
10.
Simplify . Enter your answer as a single integer or fraction.
Answer: -1
, where . When numerator and denominator are opposites, the result is .
11.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
, where and .
12.
Simplify
Answer: -(x + 3)/(x - 3)
1. Rewrite by factoring out to get standard form. -(x^2 - 9)
2. Now factor as a difference of squares. (x + 3)(x - 3)
3. Factor the denominator (perfect square). (x - 3)^2
4. Combine and simplify. What is the final answer? -(x + 3)/(x - 3)
13.
How many restrictions does have after simplifying? Enter a number.
Answer: 2
. The original denominator is zero when or . So there are 2 restrictions.
14.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
15.
Simplify
Answer: (x^2 + 2x + 4)/(x + 2)
1. Recognize the numerator pattern. is a... difference of cubes
2. Factor using . What is ? x^2 + 2x + 4
3. Factor the denominator (difference of squares). (x + 2)(x - 2)
4. What is the simplified form? (x^2 + 2x + 4)/(x + 2)
16.
Which value makes the expression undefined?
1. a)
2. b)
3. c)
4. d)
Answer:
The denominator when . Division by zero is undefined.
17.
If simplifies to , what is the value of ?
Answer: -1
. Comparing with , we get .
18.
Simplify
Answer: (x + 4)/(x - 2)
1. Factor the numerator. What two numbers multiply to 12 and add to 7? (x + 3)(x + 4)
2. Factor the denominator. What two numbers multiply to -6 and add to 1? (x + 3)(x - 2)
3. What factor cancels? x + 3
4. What is the simplified expression? (x + 4)/(x - 2)