Lesson Notes By Weeks and Term - Senior Secondary School 2
ALGEBRAIC FRACTIONS
ALGEBRAIC FRACTIONS
SUBJECT: MATHEMATICS
CLASS: SS 2
DATE:
TERM: 2nd TERM
REFERENCE BOOKS
TOPIC: ALGEBRAIC FRACTIONS CONTENT -Simplification of Algebraic Fractions. - Operation of Algebraic Fractions. SIMPLIFICATION OF ALGEBRAIC FRACTIONS. To simplify an algebraic fraction: When simplifying a fraction, remember the following facts: (x – y)2 = x2 – 2xy + y2 (Perfect Squares) X2 – 5x + 6 = x2 -2x – 3x + 6 = x(x - 2) – 3(x - 2) = (x – 2)(x – 3) Example 1 Simplify the following fractions: (a) 3x2+9x2y23x2y (b) x2-y2+3x+3yx-y+3 (c) x2-9x2+x-6 (d) 5xy-10x+y-2 8-2y2 Solution (a)3x2+9x2y23x2y = 3x2(1+3y)3x2 × y Cancel the common factors 3x2(1+3y)3x2 × y = 1+3y2y (b) x2-y2+3x+3yx-y+3 = x+yx+y+ 3(x+1)x-y+3 x+y(x-y+3)x-y+3 = x + y (c) x2-9x2+x-6 = x+3(x-3)(x+3) (x-2) = x-3x-2 (d) 5xy-10x+y-2 8-2y2 = 5xy-2+ (y-2) 2(4-y2) = y-2(5x+1) 22-y(2+y) = y-2(5x+1) 22-y(2+y) = -(5x+1) 2(2+y) Notice that in the above y – 2 = - (2 – y) In general: x – y = -(y – x) e.g. 10 – 4 = -(4 – 10) i.e. 6 = -4 + 10 6 = 6 Example 2 Simplify the following fractions: (a) x2+9x+8 x2+6x+5 (b) 6x2+30x+36 2x2+12x+16 (c) 5x2-5x-100 4x2-8x-96 (d) (6x-18y)227y2-3x2 Solution (a) x2+9x+8 x2+6x+5 = (x+8) (x+1) (x+5) (x+1) = x+ 8 x+5 (b) 6x2+30x+36 2x2+12x+16 = 6(x2+5x+6) 2(x2+6x+8) Now factorise the quadratic expressions inside the brackets: = 3x+3(x+2) x+4(x+2) = 3x+3(x+4) (c) 5x2-5x-100 4x2-8x-96 = 5(x2-x-20)4(x2- 2x-24) Now factorise the quadratic expressions inside the brackets: = 5x+4(x-5)4x-6(x+4) = 5(x-5) 4(x-6) (d) (6x-18y)227y2-3x2 = 6x-18y(6x-18y) 3(9y2-x2) = 36x-3y(x-3y) 33y-x(3y+x) But x – 3y = -(3y - x) = 123y-x(x-3y) 33y-x(3y+x) = 12(x-3y) 3y+x Algebraic Fractions: Simplification, Operation and Undefined Fractions. EVALUATION OPERATION OF ALGEBRAIC FRACTIONS. Multiplication and Division of Fractions Factorise fully first, then divide the numerator and denominator by any factors that they have in common. Example 1 Simplify a2+2a-3a2-16 × a+4a2+8a+15 Given expression = a+3(a-1)a-4(a+4) × a+4a+5(a+3) = a+1a-4(a+5) The answer should be left in the form given. Do not multiply out the brackets. Example 2 Simplify m2-a2m2+bm+am+ ab ÷ m2-2am+a2cm+bc To divide by a fraction, multiply by its reciprocal. Given expression =m2-a2m2+bm+am+ ab × cm+bcm2-2am+a2 =m-a(m+a)m+b(m+a) × c(m+b)m-a(m-a) = cm -a Example 3 Simplify =a2+ ab a3-2ab+b3 ÷ a+3ba+2b × ab-a a2+3ab+2b2 Given expression = a2+ ab a3-2ab+b3 × a+2ba+3b × ab-a a2+3ab+2b2 = a(a+b)a-b(a-b) × a+2ba+3b × a(b-a)a+b(a+2b) = a2a-ba+3b Notice that (a - b) divides into (b - a) to give -1. This is because -1 x (a - b) = (b - a). EVALUATION Addition and Subtraction of Fractions Example 1 Simplify 6a- 32b The denominators are a and 2b. The LCM of a and 2b is 2ab. Express each fraction with denominator of 2ab. 6a- 32b= 6 ×2ba ×2b- 3 ×a2b ×a = 12b2ab- 3a2ab = 12b-3a2ab Example 2 Simplify 2 + 6a2+2b23ab- 4a-b2b The denominators are 3ab and 2b. the LCM of 3ab and 2b is 6ab. Express each fraction in the expression with a denominator of 6ab. 2 + 6a2+2b23ab- 4a-b2b = 2 ×6ab6ab+ 2(6a2+2b2)6ab- 3a(4a-b)6ab = 12ab+12a2+4b2-12a2+3ab6ab = 15ab+4b26ab = b(15a+4b)6ab = 15a+4b6a Example 3 Simplify x+4x2-3x- x-19-x2 x+4x2-3- x-19-x2 = x+4x(x-3)- x-13-x(3+x) = x+4x(x-3)- x-1x-3(3+x) = x2+7x+12+x2-xxx-3(x+3) = 2x2+6x+12xx-3(x+3) = 2(x2+3x+6)xx-3(x+3) Notice that the sign in front of the fraction is changed since (3 – x) = -(x – 3). This give an LCM of x(x – 3)(x + 3). Example 3 Simplify 1a-3m- 2a+3m 1a-2m- 2a+3m= a+3m-2(a-2m)a-2m(a+3m) = a+3m-2a+4ma-2m(a+3m) = 7m-aa-2m(a+3m) EVALUATION Simplify the following. GENERAL EVALUATION/ REVISION QUESTIONS Simplify the following. WEEKEND ASSIGNMENT Objectives Theory Simplify the following. 1.(a) 7pq2r21pq3r (b) p-qq2-y2 (c) 1-p2p2-1 READING ASSIGNMENT New General Mathematics SSS2, pages 193-195, exercise 17b.
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