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# Rational Expressions

In Math, “Rational Numbers” are just numbers that can be
written as

fractions:

and so on…..

Rational Expression are just polynomials that are written
in the form

of a fraction.

Examples:

If a rational expression has ONLY multiplication and
division

involved, reducing it to simplest form is just a matter of cancelling

out common factors.

When a rational expression has addition or subtraction
involved,

it must be REWRITTEN into an expression with ONLY

MULTIPLICATION & DIVION in order to be reduced to

simplest form. You CANNOT CANCEL OUT common terms

if they are not FACTORS (something being multiplied).

If we factor the polynomials in the numerator &
denominator,

then we can cancel out COMMON FACTORS.

HERE!

After factoring the denominator, we should notice that this rational expression would be

UNDEFINED (have 0

denominator) if x = 4 or if x= -2.

So we must state the restrictions that x ? 4 and x? -2

together. You can now cancel the common factors.

**Simplify:**

How do we factor an expression when the coefficient of x2
is negative?

It’s easier to do if you factor out a -1 from the expression.

Example 1C

**Simplify:**

**Multiplying Rational Expressions**

This works the same as multiplying fractions, but make
sure

to only cancel out FACTORS from top to bottom, not side

to side. Of course each expression must be FACTORED

before you can cancel out FACTORS.

**Dividing Rational Expressions**

The way to divide rational expressions is the same method
as

dividing fractions. Just take the reciprocal of the divisor (the

rational expression after the division sign, รท ) and multiply.

First, factor the polynomials wherever possible.

Notice the (y-3x) and the (3x - y). We’ve seen this
before.

Just change one expression to look like the other and multiply by -1.

(3x-y)=-1(y-3x)