PMR Form 3 Mathematics Algebraic Fractions

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PMR Form 3 Mathematics Algebraic Fractions

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PMR Form 3 Mathematics Algebraic Fractions

  1. 1. Expanding single brackets Remember to multiply all the terms inside the bracket by the term x immediately in front of the bracket 4(2a + 3) = 8a + 12 If there is no term in front of the bracket, x multiply by 1 or -1 Expand these brackets and simplify wherever possible: 1. 3(a - 4) = 7. 4r(2r + 3) = 2. 6(2c + 5) = 8. - (4a + 2) = 3. -2(d + g) = 9. 8 - 2(t + 5) = 4. c(d + 4) = 10. 2(2a + 4) + 4(3a + 6) = 5. -5(2a - 3) = 11. 2p(3p + 2) - 5(2p - 1) = 6. a(a - 6) =
  2. 2. Expanding double brackets Split the double brackets into 2 single brackets and then expand each bracket and simplify (3a + 4)(2a – 5) “3a lots of 2a – 5 and 4 lots of 2a – 5”= 6a2 – 15a + 8a – 20 If a single bracket is squared (a + 5)2 change it into double= 6a2 – 7a – 20 brackets (a + 5)(a + 5)Expand these brackets and simplify :1. (c + 2)(c + 6) = 4. (p + 2)(7p – 3) =2. (2a + 1)(3a – 4) = 5. (c + 7)2 =3. (3a – 4)(5a + 7) = 6. (4g – 1)2 =
  3. 3. Factorising – common factors Factorising is basically the Factorising reverse of expanding brackets. Instead of removing brackets5x2 + 10xy = 5x(x + 2y) you are putting them in and placing all the common factors in front. ExpandingFactorise the following (and check by expanding): 15 – 3x =  10pq + 2p = 2a + 10 =  20xy – 16x = ab – 5a =  24ab + 16a2 = a2 + 6a =  r2 + 2 r = 8x2 – 4x =  3a2 – 9a3 =
  4. 4. Factorising – quadratic expressions a 2  2ab  b 2  a 2  b2 a 2  2ab  b 2  ax  by  cx  dy 
  5. 5. Factorising – grouping and difference of two squaresGrouping into pairs Difference of two squaresFully factorise this expression: Fully factorise this expression:6ab + 9ad – 2bc – 3cd 4x2 – 25Factorise in 2 parts Look for 2 square numbers separated by a minus. Simply Use the square root of eachRewrite as double brackets and a “+” and a “–” to get: Fully factorise these: Fully factorise these: (a) wx + xz + wy + yz (a) 81x2 – 1 (b) 2wx – 2xz – wy + yz (b) 4 – t2 (c) 8fh – 20fm + 6gh – 15gm (c) 16y2 - 64 Answers: Answers: (a) (x + y)(w + z) (a) (9x + 1)(9x – 1) (b) (2x – y)(w – z) (b) (2 + t)(2 – t) (c) (4f + 3g)(2h – 5m) (c) 16(y2 + 4)
  6. 6. Factorise each of the followingx  6m  9 2 3r  6rp  3 p 2 2 25 x  120 xy  100 y 2 2
  7. 7. Factorise each of the following( x  5)  9 2 4(m  1) 2  25
  8. 8. Simplifying Algebraic Fractions Reduce this fraction 12 43 Factorise the numerator and  denominator, cancel the 53 common factors 15Simplify by factorising1 Cancel the common factors 6c 2 32 c c  2c 2c Factorise first before cancelling
  9. 9. Simplify by factorising 5 6 p 4m6   15m 5 15 p m
  10. 10. Simplify Cancel the common factors Write down what’s left2 7(c  1) 7(c  1)  (c  1)2 (c  1)(c  1) Let’s do one that isn’t already factorised3 Cancel the common factors write down what’s left 2 x  10 2( x  5)  Grade A 3 x  15 3( x  5) factorise the numerator first factorise the denominator
  11. 11. Factorise and Simplify Cancel the common factors Write down what’s left4 ( x2  9)  ( x  3)( x  3) Grade A* x2  7 x  12 ( x  3)( x  4) Factorise the numerator first Factorise the denominatorSimplify the expressions fully1) 2x  6 4) x2  16 2x x2  4 x2) 5 x  10 5) 2 x2  8 3x  6 x2  6 x  8 x2  2 x x2  5 x  63) 6) 8 x  16 x2  x  6
  12. 12. Check your answers!!1) 2 x  6  2( x  3)  x3 x Factorise the numerator 2x 2x 5 x  10  5( x  2)2) 5 Factorise the denominator  3x  6 3( x  2) 3 Cancel the common factors Write down what’s left x2  2 x x( x  2) x3)   8 x  16 8( x  2) 24) x2  16 ( x  4)( x  4)   x4 x  4x 2 x ( x  4)5) 2 x2  8 2( x 2  4 ) 2( x  2)( x  2) 2( x  2)    x2  6 x  8 ( x  4)( x  2) ( x  4)( x  2) x4 x2  5 x  6 ( x  3)( x  2)  x26)  x2 x2  x  6 ( x  3)( x  2)
  13. 13. Factorise the numerator first Factorise and Simplify Factorise the denominatork 2  36 (a  b) 2  9b 2 Cancel the common factors(k  6) 2 a 2  2ab Write down what’s left
  14. 14. Algebraic fractions – Addition and subtraction Like ordinary fractions you can only add or subtract algebraic fractions if their denominators are the same Simplify 3 + 4_ x y Addition a c ad  bc   b d bd Multiply the top and bottom of each fraction by the same amount
  15. 15. Test yourself!!Simplify. x 3x1)  = 10 10 5 22)  = 7x 7x 4m 2m  13)  = 5 5
  16. 16. Simplify x – x-5 2 6
  17. 17. Simplify Addition1 3 a c ad  bc   x 2 x b d bd
  18. 18. Simplify Addition 3 2 a c ad  bc    x y yx b d bd
  19. 19. Simplify Subtraction 2 3p a c ad  bc 2  2  p qr pr b d bd
  20. 20. Simplify Subtraction 2 x 3x  3  a c ad  bc 2 4   b d bd
  21. 21. Algebraic fractions – Multiplication and division Again just use normal fractions principles Simplify: x x 5y 2  5t  2 y 10 y 4 x a c ac   b d bd Multiplication
  22. 22. Simplify Multiplication 2x  y 5x  a c ac y 3x  y   b d bd
  23. 23. Test yourself!!!2x 9 y 5a 2b 3c  2 3 10 x 6c 10 ab 3y a 2 5 4b c
  24. 24. Test yourself!!!x  4 6x2 2 9x  4 y 2 2 3y   3x x2 6y 3 6x  4 y
  25. 25. Simplify Division2x 4 y  a c ad7 14   b d bc
  26. 26. Simplifyx y 2 2 4( x  y ) 4  3y 9x3 y 2
  27. 27. Simplifyx2  y2 2x  2 y 3  3 xy x2 y

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