PMR Form 3 Mathematics Algebraic Fractions
58 likes24,874 views
The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.
PMR Form 3 Mathematics Algebraic Fractions
- 2. 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) =
- 3. 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 =
- 4. Factorising – common factors Factorising is basically the Factorising reverse of expanding brackets. Instead of removing brackets 5x2 + 10xy = 5x(x + 2y) you are putting them in and placing all the common factors in front. Expanding Factorise 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 =
- 5. Factorising – quadratic expressions a 2 2ab b 2 a 2 b2 a 2 2ab b 2 ax by cx dy
- 6. Factorising – grouping and difference of two squares Grouping into pairs Difference of two squares Fully factorise this expression: Fully factorise this expression: 6ab + 9ad – 2bc – 3cd 4x2 – 25 Factorise in 2 parts Look for 2 square numbers separated by a minus. Simply Use the square root of each Rewrite 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)
- 7. Factorise each of the following x 6m 9 2 3r 6rp 3 p 2 2 25 x 120 xy 100 y 2 2
- 8. Factorise each of the following ( x 5) 9 2 4(m 1) 2 25
- 9. Simplifying Algebraic Fractions Reduce this fraction 12 43 Factorise the numerator and denominator, cancel the 53 common factors 15 Simplify by factorising 1 Cancel the common factors 6c 2 32 c c 2c 2c Factorise first before cancelling
- 10. Simplify by factorising 5 6 p 4m6 15m 5 15 p m
- 11. Simplify Cancel the common factors Write down what’s left 2 7(c 1) 7(c 1) (c 1)2 (c 1)(c 1) Let’s do one that isn’t already factorised 3 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
- 12. Factorise and Simplify Cancel the common factors Write down what’s left 4 ( x2 9) ( x 3)( x 3) Grade A* x2 7 x 12 ( x 3)( x 4) Factorise the numerator first Factorise the denominator Simplify the expressions fully 1) 2x 6 4) x2 16 2x x2 4 x 2) 5 x 10 5) 2 x2 8 3x 6 x2 6 x 8 x2 2 x x2 5 x 6 3) 6) 8 x 16 x2 x 6
- 13. Check your answers!! 1) 2 x 6 2( x 3) x3 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) x 3) 8 x 16 8( x 2) 2 4) x2 16 ( x 4)( x 4) x4 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) x4 x2 5 x 6 ( x 3)( x 2) x2 6) x2 x2 x 6 ( x 3)( x 2)
- 14. Factorise the numerator first Factorise and Simplify Factorise the denominator k 2 36 (a b) 2 9b 2 Cancel the common factors (k 6) 2 a 2 2ab Write down what’s left
- 15. 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
- 16. Test yourself!! Simplify. x 3x 1) = 10 10 5 2 2) = 7x 7x 4m 2m 1 3) = 5 5
- 17. Simplify x – x-5 2 6
- 18. Simplify Addition 1 3 a c ad bc x 2 x b d bd
- 19. Simplify Addition 3 2 a c ad bc x y yx b d bd
- 20. Simplify Subtraction 2 3p a c ad bc 2 2 p qr pr b d bd
- 21. Simplify Subtraction 2 x 3x 3 a c ad bc 2 4 b d bd
- 22. 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
- 23. Simplify Multiplication 2x y 5x a c ac y 3x y b d bd
- 24. Test yourself!!! 2x 9 y 5a 2b 3c 2 3 10 x 6c 10 ab 3y a 2 5 4b c
- 25. Test yourself!!! x 4 6x 2 2 9x 4 y 2 2 3y 3x x2 6y 3 6x 4 y
- 26. Simplify Division 2x 4 y a c ad 7 14 b d bc
- 27. Simplify x y 2 2 4( x y ) 4 3y 9x3 y 2
- 28. Simplify x2 y2 2x 2 y 3 3 xy x2 y