1 P.5 RATIONAL EXPRESSIOS Objectives:  Simplify Rational Expression  Operations on Rational Expressions  Determining the LCD of Rational Expressions  Complex Fractions ( التعبير الكسرية )

2 Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are polynomials. Domain of a rational expression – the set of all real numbers except those for which the denominator is zero. Reduced form – a rational expression in which the numerator and denominator have no factors in common.

3 Rational Function: Let p(x) and q(x) be polynomials. Then a rational function is given by: The domain (values x can take) of the fcn includes all x- values such that q(x)  0.

4 Ex: Find the domain : The domain of f is all values of x, … but x  0; solve by factoring …  (x - 2)(x + 2)  0;  By ZPP, x-2  0; x+2  0  x   2  Domain=

5 solve: x  0 x 2  -1 No real solutions …. There are no restrictions on the values x can take.

6 Rational Expressions P, Q, R, and S are polynomials Addition Operation Multiplication Subtraction Division Notice the common denominator Find the reciprocal and multiply

7 Cancellation Properties

8 Rational Expressions Ex: Simplify Cancel common factors Factor Ex: Simplify Factor Cancel common factors Multiply Across

9 Factor the numerator and denominator Divide out the common factors. Write in simplified form. Ex: Simplify

10 Rewrite the problem as multiplication. Factor each rational expression Divide out the common factors. Write in simplified form.

11 Least Common Denominator( LCD) ( المقام المشترك الاصغر )

12 Adding/Subtracting Combine like terms Must have LCD: x(x + 4) Distribute and combine fractions

13 Complex Fractions Factor to get here Distribute and reduce to get here Multiply by the LCD: x ( الكسور المركبة )

14 Ex: Simplify