Unit 1: Special Products
Unit 2: Factorization
Unit 3: Triangles
Unit 4: Particular triangles and solids
Unit 1
Special products
Special Products
Product of sums and difference :
(a + b)(a - b) = a2 - ab + ab - b2 = a2 - b2
Square of a binomial
(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2
(a - b)2 = (a - b)(a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2
Square of a trinomial
(a + b + c )2= a2 + b2 + c2 + 2ab + 2ac +2bc
Cube of a binomial
(a + b)3 = (a + b)2(a + b) = (a2 + 2ab + b2)(a + b) = a3+ 3a2 b + 3ab2 + b3
(a - b)3 = (a - b)2(a - b) = (a2 - 2ab + b2)(a - b) = a3- 3a2 b + 3ab2 - b3
Binomial Expansion
For any value of n, whether positive, negative, integer or non-integer,
the value of the nth power of a binomial is given by:
Unit 2
Factorization
a2 b2 = (a + b) (a - b)
a2 + b2
No factorization
a3- b3 = (a - b) (a2+ ab + b2)
a3+ b3= (a + b) (a2- ab + b2)
a4 b4 = (a2+ b2) (a2- b2) = (a2+ b2) (a + b) (a - b)
a4+ b4
No factorization
…
...
a6- b6 = (a3+ b3) (a3- b3) = (a + b) (a2- ab + b2) (a - b) (a2+ ab + b2)
…
...
FACTORIZATION
Expressing polynomials as product of other polynomials that cannot
be further factorized is called Factorization
Example: common monomial factor
Factoring polynomials with a common monomial factor (using GCF). Always look for a
greatest common factor before using any other factoring method.
Steps:
1. Find the greatest common factor (GCF).
2. Divide the polynomial by the GCF. The quotient is the other factor.
3. Express the polynomial as the product of the quotient and the GCF.
Example :
3
2
6c d 12c d
GCF
3cd
2
3cd
Step 1:
Step 2: Divide by GCF
3
2
(6c d 12c d
2
4cd 1
3cd(2c
2
4cd 1)
3cd) 3cd
The answer should look like this:
3
2 2
Ex: 6c d 12c d
2c
2
3cd
A factoring Technique
Factoring by grouping
for polynomials
with 4 or more terms
Factoring By Grouping
1. Group the first set of terms and last set of terms with parentheses.
2. Factor out the GCF from each group so that both sets of parentheses
contain the same factors.
3. Factor out the GCF again (the GCF is the factor from step 2).
Example 1:
b
3
3b
2
4b
Step 1: Group
b
3
3b
2
4b 12
Step 2: Factor out GCF from each group
2
b b 3
4b 3
Step 3: Factor out GCF again
b
2
3 b
4
12
3
2
2 x 16 x
3
2
2 x 8x 4x
Example 2:
2
2
2
2
3
2
8x
32
64
x 8x
4x 32
2
x x 8
4 x 8
2
x 8 x 4
x 8 x 2 x 2
Observation
If the factorization is correct the degree of starting polynomial is always equal to
the sum of the degrees of each obtained factor
Examples
3x5 - 3x =
3x(x4 - 1) =
3x(x2 + 1)(x2 - 1) =
3x(x2 + 1)(x + 1)(x - 1)
V degree
I + IV degree
I + II + II degree
I + II + I + I degree
Unit 3
Triagles
Triangles
Sum of internal angles = 180
Sum of external angles =360
Relation between sides of a triangles
•The sum of the measures of two sides of a triangle is greater than
the measure of the third side
Altitude: An altitude of a triangle is a straight line through a vertex and
perpendicular to (i.e. forming a right angle with) the opposite side.
Median: A median of a triangle is a straight line through a vertex and
the midpoint of the opposite side, and divides the triangle into two equal
areas.
Bisector: An angle bisector of a triangle is a straight line through a
vertex which cuts the corresponding angle in half.
The intersection of the altitudes is the orthocenter
Internal
acute angle
External obtuse angle
The intersection of the medians is the centroid or
geometric barycenter
The centroid cuts every median in the ratio 2:1, i.e. the distance between
a vertex and the centroid is twice the distance between the centroid and
the midpoint of the opposite side.
A
G
M
AG=2/3 AM
GM=1/3 AM
In a right triangle the hypotenuse ‘s
median is equal to half of the hypotenuse
length.
Incentre: The point where the three angle bisectors of a triangle
meet.
Circumcenter: The circumcenter of a triangle is the point where
the perpendicular bisectors of the sides intersect. It is also the
center of the circumcicle, the circle that passes through all three
vertices of the triangle.
Internal acute angle
External obtuse angle
Equivalence
Two triangles are said to be equal if they have equal:
Two sides and the
common angle
A side and two angles
(II)
(I)
Similarity
Similar triangles have:
• all their angles equal
• corresponding sides have the same ratio
Equivalence
Triangles with the same base and
same altitude are equivalent
Three sides
(III)
Similarity and congruence
h
2h
L
2L
When two triangles are similar, the reduced ratio of any two
corresponding sides is called the scale factor of the similar triangles
Theorem : If two similar triangles have a scale factor of a : b, then the
ratio of their perimeters is a : b.
Unit 4
Particular triangles and solids
Equilateral triangle
Equilateral triangle inscribed
Equilateral triangle
r 3
30
L
60
L* 3
2
60
L*1/2
L = r * sen 60°
R
r
In an equilateral triangle the four particular points are coincident.
Therefore:
r=1/3h
R=2/3h
Isosceles right triangle
Square inscribed
45°
45
L 2
r 2
2r
45
L
Quadrilaterals
Sum of internal angles: 360
Sum of external angles: 360
Parallelogram: quadrilateral with two
pairs of parallel sides.
Properties
• Two pairs of opposite sides are equal in length.
• Two pairs of opposite angles are equal in measure.
• The diagonals bisect each other.
Trapezoid
Trapezoid: a quadrilateral with at
least one pair of parallel sides
Trapezio isoscele
Two triangles are equal and
two triangles are similar
Circumference theorem
The angle formed at the centre of the circle by lines originating from two
points on the circle's circumference is double the angle formed on the
circumference of the circle by lines originating from the same points.
2
2
Volume of the most important solids
Right parallelepiped
Prism
Cilinder
V=a*b*c
V=B*h
V= r2h
Piramid
Cone
Sphere
V=B*h/3
V= r2h/3
V=(4/3) r3
S=4 r2