lesson 5.1 rational expressions |

In common terms,

Look at the first five letters in the word rational. They spell __ratio__ and that's exactly what rational numbers are.

** definition **The set of

the set of all numbers represented by

where

Now, let's discuss the issue of b = 0. Our number system is constructed such that **division by zero is undefined and/or equal to infinity**. This is because of the fact that the smaller the denominator or divisor, the larger the quotient.

When we divide 8 by 4, we get 2. When we divide 8 by 2, we get 4. If we divide 8 by 0.000002, we get 4,000,000, so imagine how large the answer will be when we divide 8 by 0.

Really, zero is just a place holder half way between – 1 and 1. This is why we define a rational number to be a fraction whose denominator is not equal to zero. Thus, whenever we are working with rational numbers in algebra, we must specify that the denominator is not equal to zero.

For instance, say we are asked to solve . The very first thing we will specify in our solution is that *x* 5 since *x* = 5 would make our denominator equal to zero.

Any **integer** (whole number) such as 5, is a fraction because we can write it as . This often causes confusion for students who are doing a question such as: , which of course really means you should multiply the numerator by 2 and the denominator by 1 like this: .

**The denominator of an integer is 1.**

With **rational numbers or expressions** just follow the rules and you'll get the right answers.

multiplication & division |
change of sign |
examples |
practice |
solutions |

__multiplication____ ____and division of rational numbers__

**Fraction facts about multiplication and division**

b) | |

e) | f) |

g) |

Let's discuss each fact in the list:

a) means that a single negative can be placed anywhere in the fraction.

b) means that the negative exponent, flips the fraction.

c) means that integers have a denominator of 1.

d) tells us how to multiply fractions -- multiply the tops and multiply the bottoms.

e) means that 0 multiplying anything except an indeterminate form will = 0.

f) means that division by 0 yields a huge value, which we call infinity or undefined.

g) means that division by a fraction is the same as multiplication by its reciprocal.

**This is not only true for fractions.** This is true for all numbers.

If you want to divide something by 2, you take half of it!

Since **division is the inverse of multiplication**, dividing by **a** is the same as multiplying by , no matter what kind of number **"a**" is.

__Equivalent Fractions or ____Reducing__

Reducing (once referred to as canceling) is one of the least understood operations in math.

The most common error is reducing across addition or subtraction.

For example, many students will ** incorrectly** perform the following operation:

because they ** incorrectly** reduce or cancel the

If we do this example with integers, it becomes obvious that the result is not at all accurate

since we know that

.

So what is reducing?

Reducing, a technique used to simplify fractions in order to express them in lowest terms, is simply the act of rewriting a fraction as an equivalent value because there is **a factor of 1** in the original expression of the fraction.

For example

As you can see from the last example, the simplification of fractions or rational numbers is easily accomplished when both numerator and denominator are in factored form.

Be kind to yourself as you work with rational numbers. If your first step is to factor wherever possible, you'll have a lot less grief from fractions. Remember that reducing requires there be the same factor in a numerator and denominator.

intro |
change of sign |
examples |
practice |
solutions |

a) | Factor. |

Reduce like terms in numerator and denominator. | |

Multiply. | |

b) | |

Factor, Reduce like terms, | |

Multiply. | |

c) | |

Invert the 2nd fraction.Factor. | |

, x – y |
Reduce like terms and multiply. |

d) | |

Invert the 3rd fraction. | |

Factor. Reduce and multiply. |

intro |
multiplication & division |
change of sign |
practice |
solutions |

To illustrate this process, let's look at an example.

Reduce

First, we factor both the numerator and the denominator to get

The problem now is that (*x* – 5) is not equal to (5 – *x*), so we can't reduce.

However, if we **factor** –** 1** from the (5 – *x*), we will get – (*x* – 5), and then we can reduce.

The problem therefore can be solved by this process as shown.

.

Though we call this **change of sign**, we're not really changing any signs or values. We're simply expressing the terms in a different form to suit our purpose.

We're using the fact that a – x = – (x – a).

Perform the following operations

a) | |

Factor -1 out of denominators. Factor the trinomial. | |

Reduce and multiply | |

b) | Invert the divisor.Factor the denominator. Factor -1 in numerator and denominator of the 1st fraction. |

Factor the numerator of the 2nd fraction. | |

Reduce. Multiply | |

intro |
multiplication & division |
examples |
practice |
solutions |

Perform the indicated operations. Reduce to lowest terms.

a)

b)

c)

d)

e)

f)

g)

intro |
multiplication & division |
change of sign |
examples |
solutions |

a) |

b) = |

c) = |

d) |

e) |

f) |

g) |

intro |
multiplication & division |
change of sign |
examples |
practice |

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