1. RATIONAL EXPRESSIONS

2. Definition of a Rational Expression A rational number is defined as the ratio of two integers, where q ≠ 0 Examples of rational numbers: A rational expression is defined as the ratio of two polynomials, where q ≠ 0. Examples of rational expressions:

3. Domain of a Rational Expression The domain of a Rational Expression is the set of all real numbers that when substituted into the Expression produces a real number. If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is: {x | x  2}. illegal if this is zero Note: There is nothing wrong with the top = 0 just means the fraction = 0

4. Finding the Domain of Rational Expression Set the denominator equal to zero. The equation is quadratic. Factor the equation Set each factor equal to zero. Solve The domain is the set of real numbers except 5 and -5 Domain: {a | a is a real number and a ≠ 5, a ≠ -5}

5. REDUCING RATIONAL EXPRESSIONS To reduce this rational expression, first factor the numerator and the denominator. To find the domain restrictions, set the denominator equal to zero. The equation is quadratic. Set each factor equal to 0. p = -7 or p = 7 The domain is all real numbers except -7 and 7.

6. CAUTION: Remember when you have more than one term, you cannot cancel with one term. You can cancel factors only. REDUCING RATIONAL EXPRESSIONS There is a common factor so we can reduce. Provided p ≠ 7 and p ≠ -7

7. To reduce this rational expression, first factor the numerator and the denominator. CAUTION: Remember when you have more than one term, you cannot cancel with one term. You can cancel factors only. REDUCING RATIONAL EXPRESSIONS There is a common factor so we can reduce. 1

8. Simplifying a Ratio of -1 The ratio of a number and its opposite is -1 factor out a -1

9. Simplifying Rational Expressions to Lowest Terms. To reduce this rational expression, first factor the numerator and the denominator. Reduce common factors to lowest terms. Notice that (c – d) and (d – c) are opposites and form a ratio of -1 Solution

10. Simplifying Rational Expressions to Lowest Terms. To reduce this rational expression, first factor the numerator and the denominator. Reduce common factors to lowest terms. Notice that (y - 5) and (5 – y) are opposites and form a ratio of -1 Solution

11. Multiplication of Rational Expressions Let p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,

12. To multiply rational expressions we multiply the numerators and then the denominators. However, if we can reduce, we’ll want to do that before combining so we’ll again factor first. MULTIPLYING RATIONAL EXPRESSIONS Now cancel any like factors on top with any like factors on bottom. Simplify. 1 1

13. Division of Rational Expressions Let p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,

14. To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem. DIVIDING RATIONAL EXPRESSIONS Multiply by reciprocal of bottom fraction. 1

15. To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem. DIVIDING RATIONAL EXPRESSIONS Multiply by reciprocal of bottom fraction. Factor Notice that (p - 6) and (6 – p) are opposites and form a ratio of -1 Reduce common factors Solution

16. Addition and Subtraction of Rational Expressions Let p, q, and r represent polynomials where q ≠ 0. Then, 1. 2. Addition and Subtraction of Rational Expressions Let p, q, and r represent polynomials where q ≠ 0. Then, 1. 2.

17. Adding and Subtracting Rational Expressions with a Common Denominator The fractions have the same denominator. Add term in the numerators, and write the result over the common denominator. Simplify to lowest terms. Solution

18. Adding and Subtracting Rational Expressions with a Common Denominator The fractions have the same denominator. Subtract the terms in the numerators, and write the result over the common denominator. Simplify the numerator. Factor the numerator and denominator to determine if the rational expression can be simplified. Simplify to lowest terms.

19. Steps to Add or Subtract Rational Expressions Factor the denominators of each rational expression. Identify the LCD Rewrite each rational expression as an equivalent expression with the LCD as its denominator. Add or subtract the numerators, and write the result over the common denominator. Simplify to lowest terms.

20. To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. ADDING RATIONAL EXPRESSIONS So the common denominator needs each of these factors. This fraction needs (x + 5) This fraction needs nothing simplify distribute Reduce common factors 1 Solution

21. To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. ADDING RATIONAL EXPRESSIONS The expressions d - 7 and 7- d are opposites and differ by a factor of -1 Therefore, multiply the numerator and denominator of either expression by -1 to obtain a common denominator. Simplify Solution Add the terms in the numerators, and write the result over the common denominator.

22. To subtract rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. SUBTRACTING RATIONAL EXPRESSIONS So the common denominator needs each of these factors.The LCD is 6. This fraction needs (2) This fraction needs (3) simplify distribute Reduce common factors Solution

23. Subtracting rational expressions is much like adding, you must have a common denominator. The important thing to remember is that you must subtract each term of the second rational function. SUBTRACTING RATIONAL EXPRESSIONS So a common denominator needs each of these factors. This fraction needs (x + 2) This fraction needs (x + 6) Distribute the negative to each term. FOIL