-- Unit4 H Seconddry 3 Unit 4: Rational Functions • 4.1 -Simplifying, Multiplying and Dividing Rational Expressions --- In the previous unit we studied polynomial functions. In this unit we will expand and learn about rational functions. A rational function is defined as follows: R(x) = ~~=~, Where N(x) and D(x) are polynomial functions and D(x) '# 0 ***The Domain of a rational function is all real numbers except x-values that make the denominator zero*** ~ We have already seen some specific types of rational function. Linear, quadratic, cubic, and higher order polynomial functions are types of rational functions. -. Example 1: Determine whether each function is a rational function or is not a rational function. If it is not rational, explain why. a. h(x) = x2 - 6x - 1 b. rex)= l lhl Gr 2 c. g(x) [/1 1 d. p (x)= x2+2 x +1 ~ \~ -Y\\)\ = xi - ~1 vw\ (}. ~\~ ~()\~ Example 2: Determine the restriction(s) ofthe domain for each variable ofthe expression, then simplify. 3xb1., a.- 27xb.. b. _2:,_ 3x-15 ;;>\. t-C ') c. x -5 .l(l=2-X s ) d. x 2 - 7x+12 x 2 + 3x - 18 ( 'f --t.f)( '1.-- ?J) (y:-t~) ()(-~) -2()(-c,) 83 Example 3: Find the domain, then simplify. a. 2X2-!f x-S c. b.-x-2 10 ~)' (- <J) \) \ zsr-9 Sx 2 -12x-9 o(;l ® Remember you need to always list the restrictions before you reduce!! And, you can never reduce (simply) through addition or subtraction. 2+5 2 - therefore --:;t:) Example: x+4 x-3 4 !!! --:;t:- -3 When multiplying and dividing rational expressions you will be using the same rules that apply to multiplying and dividing rational numbers (fractions). Remember that when you multiply rational numbers, you can simplify at the beginning or the end, and the product is the same. I i I Rational Numbers _g_ Method A ..4_ . 5x2 = 1 Ox3 120x2 15x2 8 _ 1x - 1 -12 I Method B §_ = _1Q_ 120 8 0 15 Rational Expressions Involving Variables ~ 3 1 1 1 0 $= -12 112 4 1 2x . firX'l ~ ,B 3 =~ 12 4 ***Remember when multiplying or dividing always list the restrictions*** When multiplying rational expressions: 1. 2. 3. Factor all polynomials. Give restrictions. Simplify the factors. Make sure you are simplifying correctly! 84 H Secondary Unit4 ~ Example 4: Multiply the rational expressions and simplify all answers completely. -l~ Ji_J Restrictions: C. Restrictions: X f- 0 1"3 Restrictions: f -=/ -C? 1 0 1t \ x+S x- 3 ~ 2 4X+3 ~ ('f.. -3i.Y-- 9 l\~ l"SJ Restrictions: 'f1 ~ 5\ \ 13 85 H Secondary 3 Unit4 When dividing rational expressions: 1. Factor all polynomials and give restrictions. 2. Multiply by the reciprocal (to divide the fractions) and give any n~ restrictions. 3. Simplify factors. - Example 5: Divide the rational expressions and simplify all answers completely. a. ~bz + ~ • 4c SrA.Q .J&f 'L"'J.-?1~c 'l- Restrictions: 0\.:f 0 1'r; i 0 1 (, 4 0 Restrictions: 'f.1 t 3 10 -..._ _,_ x 2 -4 _,_ x - 2 x2y2-xy · 3x2+19x-14 · xy 4x d. l\ 'f. l ~~l)\3~2) . ~ :t1~~'j-\)• (Hl-}:~--2.) (}-"~ 3~~"4)L~---4) c;)'l~~~) L\1l).tt)( ~X/~ '2. hy-\')l'f..t~L~-bJ Restrictions: 1 t '-\ Y,. 1 0 \ .C? J~ Restrictions: 86

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