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