Expanding and Simplifying Algebraic Expressions

1. Expanding and Simplifying Algebraic Expressions Lesson Aims: • To be able to simplify algebraic expressions with adding, subtracting and multiplying • To be able to expand a single bracket, including negative numbers

2. Review of Algebraic Expressions So far we have learned that: 2c means 2 multiplied by c z means z divided by six. 6

3. What is the value of this expression? y2 + 5 when y = 4

4. What is the value of this expression? y2 + 5 when y = 4 (4 x 4) + 5 = 16 +5 = 21

5. Is this true for any number?a + b = b + a(hint: does 4 + 3 = 3 + 4? Imagine that a and b are numbers. Does it matter what order we use to add them?

6. Is this true for any number?a - b = b - a(hint: does 7 - 5 = 5 - 7? Imagine that a and b are numbers. Does it matter what order we use to subtract them?

7. Simplifying Expressions • What we already know • When we have • a + a + a = • We collect them together = 3a • a + 3a – a = • 1 + 3 – 1 = 3 • = 3a

8. Simplifying Expressions • What we already know • a x a x a x a = similar to indices • 4 x 4 x 4 x 4 = 44 • So a x a x a x a = a4

9. Simplifying Expressions 2p + 4p = +

10. Simplifying Expressions 2p + 4p = So 2p + 4p = 6p +

11. Simplifying Expressions 2a + 3a = +

12. Simplifying Expressions 2a + 3a = + = 2a +3a = 5a

13. Simplifying Expressions 2a + p = +

14. Simplifying Expressions We can only simplify when the terms have the same letter or variable. 2a + p = + = 2a + p

15. Simplifying Expressions 2a + p + 4a + 2k + 3p =

16. Simplifying Expressions 2a + p + 4a + 2k + 3p = First look at a: Then look at p: Then look at k:

17. Simplifying Expressions 2a + p + 4a + 2k + 3p = First look at a: 2a + 4a = 6a Then look at p: Then look at k:

18. Simplifying Expressions 2a + p + 4a + 2k + 3p = First look at a: 2a + 4a = 6a Then look at p: p + 3p = 4p Then look at k:

19. Simplifying Expressions 2a + p + 4a + 2k + 3p = First look at a: 2a + 4a = 6a Then look at p: p + 3p = 4p Then look at k: 2k

20. Simplifying Expressions 2a + p + 4a + 2k + 3p = First look at a: 2a + 4a = 6a Then look at p: p + 3p = 4p Then look at k: 2k So the expression becomes: 6a + 4p + 2k

21. Simplifying Expressions • Try these • 4f + 7b – f + 3b = • h + 3h – 2h + 2h – 3h = • 8a + b – c – 7a = • 3y + b + 2b – y + 8 = • 4c - 2d – 2c – 3d = • extension • 8d³ + 2d² - 3d² = • 3s³ + 4s³ - s =

22. Simplifying Expressions • a x a x a x b x b = a3b2 • a x b x c = abc • 2a x b x c = 2abc d x r x d x r = r x r x d x d = r2d2

23. Simplifying Expressions • 2a x 2b x c = • Remember with multiplying no matter what order you put the numbers your multiplying you will get the same answer. • So I will put the numbers and the letters together

24. Simplifying Expressions • 2a x 2b x c = • 2 x 2 x a x b x c = 4abc

25. Simplifying Expressions • s x s x s x s = • r x s x r x s x r = • 4d x 3 d = • 2a x 3d x 4a = • 6c x 6v x 4b = • 3n x 5 b x f= • Extension • 3r x 3r x 3r =

26. Expanding Brackets 3(a + 5) What does this mean? ‘add five to a then multiply the whole lot by three’ Or ‘three lots of a added to three lots of 5

27. Expanding Brackets 3(a + 5) + 5 + 5 a a + 5 a

28. Expanding Brackets 3(a + 5) + 5 + 5 a a + 5 a 3(a + 5) =

29. Expanding Brackets 3(a + 5) + 5 + 5 a a + 5 a 3(a + 5) = (3 x a) +

30. Expanding Brackets 3(a + 5) + 5 + 5 a a + 5 a 3(a + 5) = (3 x a) + (3 x 5) =

31. Expanding Brackets 3(a + 5) + 5 + 5 a a + 5 a 3(a + 5) = (3 x a) + (3 x 5) = 3a + 15

32. Expanding Brackets 6(2a + 4) + 4 + 4 + 4 + 4 + 4 + 4 = 12a + 24 (6 x 2a) + (6 x 4) 6(2a + 4) =

33. Expanding Brackets Example: 5(2z – 3) Each term inside the brackets is multiplied by the number outside the brackets. Watch out for the signs!

34. Expanding Brackets Example: 5(2z – 3) (5 x 2z) – 5 x 3

35. Expanding Brackets Example: 5(2z – 3) (5 x 2z) – 5 x 3 = 10z – 15

36. Expanding Brackets • Example • t( 3 +7) • Rememeber muliplty everything inside the bracket by whats outside the bracket. • (t x 3) + (t x 7) • 3t + 7t • = 10t

37. Expanding Brackets • Example • r(6m -5) • (r x 6m) – (r x 5) • 6mr – 5r

38. Expanding Brackets extension y(2 + y) a(7 – 8) p(3p + q) s(6s – 5) k(2m + n) • 4(y + 3) • 5(a + 9) • 3(x – 7) • 6(g – 2) • 2(3b + 7) • 10(3k + 10) • 12(4s + 6)

39. Expanding Brackets When we have 2 brackets the rules are the same. Multiply what's inside the bracket be what is out of the bracket. Remember to be careful of signs. We are only going to have addition in the middle of the brackets today is minus gets tricky! Example: 2(3p + 4) + 3(4p + 1)

40. Expanding Brackets Example: 2(3p + 4) + 3(4p + 1) = (2 x 3p) + (2 x 4)

41. Expanding Brackets Example: 2(3p + 4) + 3(4p + 1) = (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1)

42. Expanding Brackets Example: 2(3p + 4) + 3(4p + 1) = (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1) = 6p + 8

43. Expanding Brackets Example: 2(3p + 4) + 3(4p + 1) = (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1) = 6p + 8 + 12p + 3

44. Expanding Brackets Example: 2(3p + 4) + 3(4p + 1) = (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1) = 6p + 8 + 12p + 3 = 18p + 11

45. Expanding Brackets • 5(x + 3) + 2(x – 7) • First bracket • 5x + 15 • Second bracket • 2x -14 • simplify • 5x + 15 + 2x -14 • 7x +1

46. Expanding Brackets • 3(x – 5) + 4(x + 6) • 3(2x – 1) +2(3x + 4) • 4(5x + 2) + 8(2x – 1) • 5(x + 4) + (x – 8) • 8(x – 1) + (3x – 4) • 3(2x – 5) + (4x + 9)