IB Math Studies – Topic 2

1. IB Math Studies – Topic 2 Number and Algebra

2. IB Course Guide Description

3. IB Course Guide Description

4. Set Language • A set is a collection of numbers or objects. - If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers. • An element is a member of a set. - 1,2,3,4 and 5 are all elements of A. -  means ‘is an element of’ hence 4  A. -  means ‘is not an element of’ hence 7  A. -  means ‘the empty set’ or a set that contains no elements.

5. Subsets • If P and Q are sets then: • P  Q means ‘P is a subset of Q’. • Therefore every element in P is also an element in Q. For Example: {1, 2, 3} {1, 2, 3, 4, 5} or {a, c, e}  {a, b, c, d, e}

6. Union and Intersection • P Q is the union of sets P and Q meaning all elements which are in P or Q. • P ∩ Q is the intersection of P and Q meaning all elements that are in both P and Q. A = {2, 3, 4, 5} and B = {2, 4, 6} A  B = A ∩ B =

7. Number Sets Reals Rationals (fractions; decimals that repeat or terminate) Irrationals (no fractions; decimals that don’t repeat or terminate) Integers (…, -2, -1, 0, 1, 2, …) Natural (0, 1, 2, …) Counting (1, 2, …) + *

8. Number Sets • N* = {1, 2, 3, 4, …} is the set of all counting numbers. • N = {0, 1, 2, 3, 4, …} is the set of all natural numbers. • Z = {0, + 1, + 2, + 3, …} is the set of all integers. • Z+ = {1, 2, 3, 4, …} is the set of all positive numbers. • Z- = {-1, -2, -3, -4, …} is the set of all negative numbers. • Q = { p / q where p and q are integers and q ≠ 0} is the set of all rational numbers. • R = {real numbers} is the set of all real numbers. All numbers that can be placed on a number line.

9. Arithmetic Sequences

10. Arithmetic Sequences

11. Arithmetic Series

12. Geometric Sequences

13. Geometric Sequences

14. Geometric Series

15. Solving Pairs of Linear Equations Elimination 1) Choose a variable to eliminate 2) Make coefficients opposite numbers by multiplying 3) Add the equations; solve. 4) Substitute to solve for the remaining variable. Or use GDC – Graph both Equations and find Intersection

16. Solve by Substitution or Elimination x + y = 14 x – y = 4 3x – 2y = -3 3x + y = 3 3x + 2y = 2 3x + y = 7 2x + y = 9 x + 4y = 1 4x – 5y = 3 3x + 2y = -15

17. Solving Quadratic Equations - Factoring GCF • Always look for _____ first. • Two terms usually means ________________ • Three terms usually means ______________ • x2 + bx + cnormal • ax2 + bx + cHoffman Method • Check your answer by __________. difference of squares factoring trinomials multiplying

18. FACTOR 3x2 + 15x 12x – 4x2 (x – 1)2 – 3(x – 1) (x + 1)2 + 2(x + 1) = 3x(x + 5) = 4x(3 – x) = (x – 1)(x – 4) = (x + 1)(x + 3)

19. FACTOR 9x2 – 64 100a2 – 49 36 – t10 a2b4 – c6d8 a4 – 81b4 = (3x – 8)(3x + 8) = (10a + 7)(10a – 7) = (6 – t5)(6 + t5) = (ab2 – c3d4)(ab2 + c3d4) = (a2 + 9b2)(a – 3b)(a + 3b)

20. FACTOR h2 – 17h+ 66 t2+ 20t + 36 q2– 15qr + 54r2 w2– 12wx + 27x2 • w2 – 6w – 16 • u2+ 18u + 80 • x2 – 17x – 38 • y2 + y – 72 = (w – 8)(w + 2) = (h– 11)(h– 6) = (u + 8)(u + 10) = (t + 18)(t + 2) = (x– 19)(x + 2) = (q – 9r)(q – 6r) = (y + 9)(y – 8) = (w – 9x)(w – 3x)

21. FACTOR 10 + 3x – x2 32 – 14m – m2 x4 + 13x2 + 42 5m2+ 17m + 6 8m2– 5m – 3 4y2 – y – 3 4c2 + 4c – 3 6m4+ 11m2+ 3 4 + 12q + 9q2 6x2 + 71xy – 12y2 = (5 – x)(2 + x) = (y – 1)(4y + 3) = (16 + m)(2 – m) = (2c + 3)(2c– 1) = (x2 + 7)(x2 + 6) = (2m2+ 3)(3m2+ 1) = (m + 3)(5m + 2) = (2 + 3q)2 = (8m + 3)(m– 1) = (6x – y)(x + 12y)

22. FACTOR Completely 24x2 – 76x + 40 3a3 + 12a2 – 63a x3 – 8x2 + 15x 18x3 – 8x 5y5 + 135y2 2r3 + 250 3m2 – 3n2 2x2 – 12x + 18 = 4(2x– 5)(3x – 2) = 5y2(y + 3)(y2 – 3y + 9) = 2(r + 5)(r2 – 5r + 25) = 3a(a + 7)(a – 3) = x(x – 5)(x – 3) = 3(m + n)(m – n) = 2x(3x – 2)(3x + 2) = 2(x – 3)2

23. Solving Quadratic Equations – Quadratic Formula