The area of a rectangle is . The width of the rectangle is . What is the length of the rectangle?

### Guidance

Recall that a rational function is a function,
, such that
, where
and
are both polynomials. A
**
rational expression
**
, is just
. Like any fraction, a rational expression can be simplified. To simplify a rational expression, you will need to factor the polynomials, determine if any factors are the same, and then cancel out any like factors.

Fraction:

Rational Expression:

With both fractions, we broke apart the numerator and denominator into the prime factorization. Then, we canceled the common factors.

**
Important Note:
**
is completely factored.
**
Do not
**
cancel out the
’s!
reduces to
, but
does not because of the addition sign. To prove this, we will plug in a number for
to and show that the fraction does not reduce to
. If
, then
.

#### Example A

Simplify .

**
Solution:
**
The numerator factors to be
and the denominator is
.

#### Example B

Simplify .

**
Solution:
**
If you need to review factoring, see the
*
Factoring When the Leading Coefficient Equals 1
*
concept and the
*
Factoring When the Leading Coefficient Doesn't Equal 1
*
concept. Otherwise, factor the numerator and find the GCF of the denominator and cancel out the like terms.

#### Example C

Simplify .

**
Solution:
**
Factor both the top and bottom and see if there are any common factors.

**
Special Note:
**
Not every polynomial in a rational function will be factorable. Sometimes there are no common factors. When this happens, write “not factorable.”

**
Intro Problem Revisit
**

Recall that the the area of a rectangle is the length times the width. To find the length, we can therefore divide the area by the width. So we're looking for .

If we factor the numerator and the denominator, we get:

Therefore, the length of the rectangle is .

### Guided Practice

If possible, simplify the following rational functions.

1.

2.

3.

4.

#### Answers

1.

2. There are no common factors, so this is reduced.

3.

4. In this problem, the denominator will factor like a quadratic once an is pulled out of each term.

### Explore More

- Does simplify to ? Explain why or why not.
- Does simplify to ? Explain why or why not.
- In your own words, explain the difference between the previous two expressions and why one simplifies and one does not.

Simplify the following rational expressions.