Chapter 9 Rational Expressions In Math, “Rational Numbers” are just numbers that can be written as fractions: 2 = 2/1.1 = 1/ = -3 ¾ = -15/4 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. PAUSE RIGHT 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 Now this “rational expression” is in the form of factors multiplied together. You can now cancel the common factors. WRONG!

Simplify: How do we factor an expression when the coefficient of x 2 is negative? It’s easier to do if you factor out a -1 from the expression. Example 1C

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, ÷ ) 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) 1