Rational Expressions and Functions: Adding and Subtracting Objectives: When Denominators are the same When Denominators are different

When Denominators Are the Same Addition and Subtraction with Like Denominators Let p, q, and r represent polynomials where q ≠ 0. To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator 1. 2.

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

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.

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.

When Denominators Are Different Factor the denominators of each rational expression. Identify the LCD To find the least common multiple of two or more expressions: Find the prime factorization of each expression Form a product that contains each factor the greatest number of times that it occurs in any one prime factorization 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.

Find the least common multiple of each pair of polynomials Solution We write the prime factorizations of The factors 3, 7, and x must appear in the LCM if 21x is to be a factor of the LCM. The factors 3, x, and x must appear in the LCM if 3x² is to be a factor of the LCM

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

ADDING RATIONAL EXPRESSIONS To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. Simplify Add the terms in the numerators, and write the result over the common denominator. 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. Solution

SUBTRACTING RATIONAL EXPRESSIONS To subtract rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. distribute simplify This fraction needs (2) This fraction needs (3) So the common denominator needs each of these factors.The LCD is 6. Reduce common factors NONE Solution

SUBTRACTING RATIONAL EXPRESSIONS 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. FOIL Distribute the negative to each term. This fraction needs (x + 2) This fraction needs (x + 6)) So a common denominator needs each of these factors.

Find simplified form for the function given by and list all restrictions on the domain Solutions Factoring, Note that x ≠ -2, 2 Multiplying by -1/-1 since 2-x is the opposite of x-2

The LCD is (x-2)(x+2) Multiplying by 1 to get the LCD Removing a factor equal to 1: that x ≠ -2