IB Math Studies – Topic 2

Number and Algebra

IB Course Guide Description

IB Course Guide Description

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.

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}

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 =

Reals

Rationals

Integers(…, -2, -1, 0, 1, 2, …)

Natural(0, 1, 2, …)

Counting(1, 2, …)

Irrationals

Number Sets

(fractions; decimals that repeat or terminate)

(no fractions; decimals that don’t repeat or terminate)

, 2, .etc

* +

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.

Arithmetic Sequences

Arithmetic Sequences

Arithmetic Series

Geometric Sequences

Geometric Sequences

Geometric Series

Solving a System of Equations

a.k.a. “simultaneous equations”

Substitution

1) Solve one of the equations for one of the variables.

2) Substitute into the other equation

3) Solve

4) Substitute to solve for the remaining variable.

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.

Solving Pairs of Linear Equations

Or use GDC – Graph both Equations and find Intersection

Solve by Substitution or Elimination

x + y = 14x – y = 4

2x + y = 9

x + 4y = 1

3x – 2y = -3

3x + y = 3

3x + 2y = 23x + y = 7

4x – 5y = 33x + 2y = -15

• 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 __________.

Solving Quadratic Equations - Factoring

GCF

difference of squares

factoring trinomials

multiplying

FACTOR

1) 3x2 + 15x

2) 12x – 4x2

3) (x – 1)2 – 3(x – 1)

4) (x + 1)2 + 2(x + 1)= (x – 1)(x – 4)

= 3x(x + 5)

= 4x(3 – x)

= (x + 1)(x + 3)

FACTOR5) 9x2 – 64

6) 100a2 – 49

7) 36 – t10

8) a2b4 – c6d8

9) a4 – 81b4= (a2 + 9b2)(a – 3b)(a + 3b)

= (3x – 8)(3x + 8)

= (10a + 7)(10a – 7)

= (6 – t5)(6 + t5)

= (ab2 – c3d4)(ab2 + c3d4)

FACTOR10) w2 – 6w – 16

11) u2 + 18u + 80

12) x2 – 17x – 38

13) y2 + y – 72

14) h2 – 17h + 66

15) t2 + 20t + 36

16) q2 – 15qr + 54r2

17) w2 – 12wx + 27x2

= (u + 8)(u + 10)

= (x – 19)(x + 2)

= (h – 11)(h – 6)

= (t + 18)(t + 2)

= (q – 9r)(q – 6r)

= (w – 9x)(w – 3x) = (y + 9)(y – 8)

= (w – 8)(w + 2)

FACTOR18) 10 + 3x – x2

19) 32 – 14m – m2

20) x4 + 13x2 + 42

21) 5m2 + 17m + 6

22) 8m2 – 5m – 3

= (m + 3)(5m + 2)

= (8m + 3)(m – 1)

23) 4y2 – y – 3

24) 4c2 + 4c – 3

25) 6m4 + 11m2 + 3

26) 4 + 12q + 9q2

27) 6x2 + 71xy – 12y2

= (2 + 3q)2

= (5 – x)(2 + x)

= (16 + m)(2 – m)

= (2m2 + 3)(3m2 + 1)= (x2 + 7)(x2 + 6)

= (2c + 3)(2c – 1)

= (y – 1)(4y + 3)

= (6x – y)(x + 12y)

FACTOR Completely

28) 24x2 – 76x + 40

29) 3a3 + 12a2 – 63a

30) x3 – 8x2 + 15x

31) 18x3 – 8x= 2x(3x – 2)(3x + 2)

32) 5y5 + 135y2

33) 2r3 + 250

34) 3m2 – 3n2

35) 2x2 – 12x + 18

= 2(x – 3)2

= 4(2x – 5)(3x – 2)

= 3a(a + 7)(a – 3)

= 3(m + n)(m – n)= x(x – 5)(x – 3)

= 2(r + 5)(r2 – 5r + 25)

= 5y2(y + 3)(y2 – 3y + 9)

Solving Quadratic Equations – Quadratic Formula