Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression is the set of all real numbers for which v 0. Definition of a rational function Let u(x) and v(x) be polynomial functions. The function f(x) = u(x) v(x) is a rational function. The domain of f is the set of all real numbers for which v(x) 0. Ex.1 Find the domain of each rational function. (1) f(x) = 4 x x (2) g(x) = 1 x x+6 Ex.2 Find the domain of each rational function. (1) f(x) = 5x x 2 16 (2) g(x) = 3x 1 x 2 2x 3 1
Definition of a rational expression Let u, v, and w be real numbers, variables, or algebraic expressions such that v 0 and w 0. Then uw vw = u v Ex.3 Simplify the rational expression 2x 3 6x 6x 2 Ex.4 Simplify the rational expression x 2 + 2x 15 3x 9 2
Ex.5 Simplify the rational expression x 3 16x x 2 2x 8 Ex.6 Simplify the rational expression 2x 2 9x + 4 12 + x x 2 3
Ex.7 Simplify the rational expression 4x 3 y 5xy 2 2xy Ex.8 Simplify the rational expression 4x 2 y y 3 2x 2 y xy 2 4
Section 6.2: Multiplying and Dividing Rational Expressions Multiplying rational expressions Let u, v, w, and z be real numbers, variables, or algebraic expressions such that v 0 and z 0. Then u v w z = uw vz Simplify the fraction if possible. Ex.1 Multiply the rational expressions 4x 3 y 3xy 4 6x2 y 2 10x 4 Ex.2 Multiply the rational expressions x 5x 2 20x x 4 2x 2 + x 3 5
Ex.3 Multiply the rational expressions 4x 2 4x x 2 + 2x 3 x2 + x 6 4x Ex.4 Multiply the rational expressions x 2 3x + 2 x + 2 3x x 2 2x + 4 x 2 5x 6
Multiplying rational expressions Let u, v, w, and z be real numbers, variables, or algebraic expressions such that v 0, w 0, and z 0. Then u v w z = u v z w = uz vw Simplify the fraction if possible. Ex.5 Divide the rational expressions x x + 3 4 x 1 Ex.6 Divide the rational expressions 2x 3x 12 x2 2x x 2 6x + 8 7
Ex.7 Divide the rational expressions x 2 y 2 2x + 2y 2x2 3xy + y 2 6x + 2y Section 6.3: Adding and Subtracting Rational Expressions Multiplying rational expressions Let u, v, w be real numbers, variables, or algebraic expressions such that w 0. Then u w + v w = u + v w u w v w = u v w Ex.1 Combine and simplify the following rational expressions x 4 + 5 x 4 8
Ex.2 Combine and simplify the following rational expressions x x 2 2x 3 3 x 2 2x 3 Ex.3 Combine and simplify the following rational expressions x 2 26 x 5 2x + 4 x 5 + 10 + x x 5 9
Ex.4 Find the least common multiple of (1) x 2 x and 2x 2 (2) 3x 2 + 6x and x 2 + 4x + 4 (3) 5x 25 and 2x 2 9x 5 Ex.5 Add the rational expressions 7 6x + 5 8x Ex.6 Subtract the rational expressions 3 x 3 5 x + 2 10
Ex.7 Add the rational expressions 6x x 2 4 + 3 2 x Ex.8 Subtract the rational expressions x x 2 5x + 6 1 x 2 x 2 11
Ex.9 Combine and simplify the following rational expressions 4x x 2 16 + x x + 4 2 x Section 6.4: Complex Fractions Definition of complex fractions A complex fraction is a fraction that has a fraction as numerator, or denominator, or both. Ex.1 Simplify the complex fraction ( 4 18 ) ( 2 3 ) 12
Ex.2 Simplify the complex fraction ( 4y3 (5x) 2 ) ( (2y)2 10x 3 ) Ex.3 Simplify the complex fraction ( x+1 x+2 ) ( x+1 x+5 ) 13
Ex.4 Simplify the complex fraction ( x2 +4x+3 x 2 ) 2x + 6 Ex.5 Simplify the complex fraction ( x 3 + 2 3 ) (1 2 x ) 14
Ex.6 Simplify the complex fraction ( 2 x+2 ) ( 3 x+2 + 2 x ) Ex.7 Simplify the complex fraction 5 + x 2 8x 1 + x 15
Section 6.5: Dividing Polynomials Ex.1 Perform the division and simplify 12x 2 20x + 8 4x Ex.2 Divide x 2 + 2x + 4 by x 1. 16
Ex.3 Divide 13x 3 + 10x 4 + 8x 7x 2 + 4 by 3 2x. Ex.4 Divide x 3 2 by x 1. 17
Ex.5 Divide x 4 + 6x 3 + 6x 2 10x 3 by x 2 + 2x 3. 18