GONZAGA JESUIT COLLEGE, OKIJA

WEEK ONE
TOPIC: SURDS
Objectives: the students should be able to
- Understand the concept of surd and their rules
- Simplify surds in their basic form
- Perform basic operation with surds
- Rationalizing the denominator
- Solve equations involving surds

Rules of Surds
Surds are irrational numbers. They are the root of rational numbers whose value cannot
be expressed as exact fractions. Examples of surds are: √2, √7, √12, √18, etc.
1. √(a X b ) = √a X √ b
2. √(a / b ) = √a / √b
3. √(a + b ) ≠ √a + √b
4. √(a – b ) ≠ √a - √b

Basic Forms of Surds

√a is said to be in its basic form if A does not have a factor that is a perfect square.
E.g. √6, √5, √3, √2 etc. √18 is not in its basic form because it can be broken into √
(9x2) = 3√2. Hence 3√2 is now in its basic form.

Similar Surds
Surds are similar if their irrational part contains the same numerals e.g.
1. 3√n and 5√n
2. 6√2 and 7√2

Conjugate Surds
Conjugate surds are two surds whose product result is a rational number.
(i)The conjugate of √3 - √5 is √3 + √5
The conjugate of -2√7 + √3 is 2√7 - √3
In general, the conjugate of √x + √y is √x - √y
The conjugate of √x - √y = √x + √y

Simplification of Surds
Surds can be simplified either in the basic form or as a single surd.

Examples
Simplify the following in its basic form (a) √45 (b) √98

Solution
(a) √45 = √ (9 x 5) = √9 x √5 = 3√5
(b) √98 = √ (49 x 2) = √49 x √2 = 7√2

Examples
Simplify the following as a single surd (a) 2√5 (b) 17√2

Solution
(a) 2√5 = √4 x √5 = √ (4 x 5) = √20
(b) 17√2 = √289 x √2 = √ (289 x 2) = √578
Addition and Subtraction of Surds
Surds in their basic forms which are similar can be added or subtracted.

Examples
Evaluate the following
(a)√32 + 3√8 (b) 7√3 - √75 (c) 3√48 - √75 + 2√12
Solution
(a) (√32 + 3√8

= √ (16 x 2) + 3√ (4 x 2)
=4√2 + 6√2
= 10√2

(b) 7√3 - √75

= 7√3 - √ (25 x 3)
=7√3 – 5√3 =2√2

(c) 3√48 - √75 + 2√12

= 3√ (16 x 3) - √ (25 x 3) + 2√ (4 x 3)
= 12√3 - 5√3 + 4√3
= 11√3

Evaluation
1. Simplify the following (a) 5√ 12 - 3√ 18 + 4√72 + 2√75 (b) 3√2 - √32 + √50 + √98
2. Simplify the following as a single surd (i) 8√3 (ii) 13√2

Multiplication and Division of Surds

Example: Evaluate the following (a) √45 x √28 (b) √24 /√50

Solution
(a) √45 x √28
= √ (9 x 5) x √ (4 x 7)
= 3√5 x 2√7
= 3 x 2 x √ (5 x 7)
= 6√35

(b)√24 / √50
= √ (24 / 50)
= √ (12 / 25)
= √12 / √25
= √ (4 x 3) / 5
= 2√3 / 5

Evaluation:
Simplify 1. √6 x (3 - √5) 2. (2√3 - √7)(2√3 + √7)
2. Multiply the following by their conjugate (a) √3 - 2√5 (b) 3√2 + 2√3

Surds Rationalisation
Rationalisation of surds means multiplying the numerator and denominator by the
denominator or by the conjugate of the denominator.
(a) Example: Evaluate the following (a) 6/√3 (b) 3
√3 + √2
Solution
(a) 6/√3 (b) 3
= 6 x √3 √3 + √2
√3 x √3 = 3 (√3 - √2)
= 6√3 (√3 + √2) (√3 - √2)
3 = 3√3 - 3√2
= 2√3 (√3) 2 – (√2)2
= 3√3 - 3√2
3-1
= 3√3 - 3√2
1
= 3(√3 -√2)
Equality of Surds
Given two surds i.e P + √m and q + √n if P +√m = q + √n then
P - q = √n - √ m the L.H.S
Of the equation is a rational number while the L.H.S and R.H.S can only be equal if they
are both equal to zero (0)
P–q=0
:. P = q and √n - √ m = 0 i.e.
√n = √m

Examples:
Find the square root of the following?
a) 7 + 2√10 b) 14 - 4√6

Solution
(a) Let the square root of 7 + 2 √10 be √m + √n
(√m + √n)2 = 7 + 2√10
m +√2mn+ n = 7 + 2√10
m+n =7 (1)
2√mn = 2√10
mn = 10
Squaring both surds we have
mn = 10 _______(ii)
m + n = 7 ______ (i)
m n = 10 _______ (ii)
From equation (1) m = 7 – n
Put m in (ii) we have
(7 – n) n = 10
7n – n2 = 10
In sum; n2 – 7n + 10 = 0
n2 – 2n – 5n + 10 =0
n (n – 2) – 5 (n – 2) = 0
(n -5) (n – 2) = 0
n = 5 or 2
m = 7 – 2, where n = 2
m = 5,
m = 7 – 5 , when n = 5
m=2
m= 5 or 2
The square root of 7 + √10 are 5 & + 2.
(b) Let the square root of 14 – 4√6 be √P - √Q
The (√P - √Q) 2 =14 – 4√6
P - 2√PQ + Q = 14 – 4 √6
P + Q = 14 …………………… (1)

-2√PQ = - 4√6
-2 -2
√PQ = 2√6 (squaring both sides)

PQ = (2√6)2
PQ = 4 x 6 ……………………………….. (11)

P + Q = 14 ………………………………… (1)

PQ = 24 ……………………………………… (11)
From equation……………… (1) P = 14 - Q
Sub for p in equation ………………… (11)
(14 – Q) Q = 24
14Q – Q2 = 24
In turn we have:
Q2 – 14Q + 24 = 0
Q2 – 12Q – 2Q + 24 = 0
Q (Q -12) – 2 (Q – 12) = 0
Q = 2 or 12
If P = 14 – Q ,when Q= 12
P = 14 – 12
P=2
If P = 14 – Q when Q = 2
P = 14 - 2
= 12
(√12 – √2 ) = (2 √ 3 - √2) and
(√2 - √12) = (√2 - 2 √3)

Evaluation:
1. Express 3√2 - √3 in the form √m where m and n are whole number.
2√3 - √2 √n

2. Express 1 in the form p√5 + q√3, where p and q are rational numbers.
√5 +√3
WEEK 2: Polynomials

Objectives: The students should be able to

- Define and identify polynomials
- Solve polynomials using arithmetic operations
- Factorization of polynomials
- Solve remainder and factor theorem
- Find the zeros of polynomials

Polynomials are expressions with progressive powers of variables which do not include negative
or fractional powers. Examples includes,
 (i)
(ii)

Addition and Subtraction of Polynomial

WORKED EXAMPLE
Given the polynomials:
, , and
Find: (i) (ii)
Solution:

(i)

(ii) DIY.

Multiplication of Polynomials
WORKED EXAMPLES
Given that: and , find
Solution:

Division of Polynomials

The polynomial is divisible by another polynomial if the degree of is greater than or

equal to the degree of .
The Remainder Theorem:

If the polynomial is divided by , the reminder is

Worked example 1

Find the remainder when:

is divided by
Solution:
The Factor Theorem:
If by is a factor of the polynomial, , then
Worked example 2

Show that is a factor of if

Solution:

(QED) It follows that , when is divided by .

Evaluation Test

If is a factor of and leaves a remainder of 12 when it is divided

by , find:
a) The values of the constants p and q
b) The three values of x for which

Week 3: Algebraic

Equations Factorization Of Quadratic Trinomials

Objectives: the students should be able to

- Solve systems of equations

- Solve quadratic equations

- Solve problems involving symmetric properties of roots

quadratic equation.

Example:

Factorize the quadratic trinomial:

Solution:

Step 1: Find the product of the first and the last term:
Step 2: Think of two factors of such that their sum will be (The middle term), since
the middle term is negative, the lager factor will be negative:
The two factors are: ,
Step 3: Use the two factors to replace ,

=
Step 4: Group in twos and factorize as follows:

Factor out the common factor

Evaluation Test:
Factorize the following
(i) (ii)

SOLUTION TO QUADRATIC EQUATION

a) Using the Method of Completing the Square.

Example1
Solve for x in
Solution:

Isolate the constant term:

Divide all by the coefficient of the highest degree

Complete the square by adding to both sides of the equation

Take the square roots of both sides

 b) Using Quadratic Formula.

The quadratic formula is given by:

Example 1
Use the quadratic formula to solve:

Equal Roots

For a quadratic equation to have equal roots, the discriminant is equal to zero. Hence for equal
roots, we have:
Sum and Product of Quadratic Equation

For the quadratic equation: whose roots are:

Sum of roots: , Product of roots:
Example 1: If are the roots of the equation , find the value of

Solution:
Substituting values, we have:

Maximum And Minimum Values Of A Curves

In calculus, we can find the maximum and minimum value of any function without even looking
at the graph of the function. Maxima will be the highest point on the curve within the given range
and minima would be the lowest point on the curve
How to find the maximum and minimum value of a curve?

The first step for finding a minimum or maximum value is to find the critical point by setting the
first derivative equal to 0. We can then use the critical point to find the maximum or minimum
value of a parabola by plugging it back into the original function.

The parabola has a minimum if and a maximum if

Example

Find the maximum value of the function: and the coordinates of the point
where this maximum point occurred.
Solution:
, at maximum value, set first derivative equal to 0

Week 4: Logic 1

Objectives: the students should be able to

- Identify and write statements
- Recognize true and false statements
- Identify antecedents and consequents of simple statements

Statements
A proposition or statement is a verbal or written declarative sentence that is either true or false.
For example "four plus five equals nine" and "two plus two equals five are propositions or
statements. While the first statement is true, the second statement is false
The truth or falsity of a statement of proposition is called its truth value.
A true statement has a truth value T while a false statement has a truth value F. Consider the
following declarative sentences.
(a) The earth is a planet.

(b) 3+4=8.

(c) 8 {7,8,9,10}.

(d) Are you a Nigerian?

(e) x+5=9.

(f) Come out.

(g) What a great day!

The sentences: (a), (b), and (c) are propositions or statements. The first is true, the second is false
while the third is true.
(d) is a question and not a declarative sentence, hence, it is not a proposition.
Although (e) is a declarative sentence it is not a proposition because its truth or falsity depends
on the value of x.
(f) is a command and hence not a statement. (g) is an exclamation and hence not a statement.
Negation of Statement

Given that p is a proposition, the negation of p is denoted by ~p, is a proposition that is false
when p is true and true when p is false.
Example 1
Consider the proposition P : Abuja is in Nigeria
The negation of p is the proposition ~p:

It is false that Abuja is in Nigeria.

In a normal form we write:

~p: Abuja is not in Nigeria.
Note
In the true sense of the word „not’, is not a connective as it does not join two statements. It is not
a binary operation but rather a unary operation. Nevertheless ~p is a proposition provided p is
also a proposition.
Also note that the following propositions are equivalent in meanings:
s: All goats are mammals; r: every goat is a mammal; q: each

goat is a mammal; p: any goat is a mammal. The negation of

the proposition. p: All goats are mammals is any of the

following propositions.

~p: Some goats are not mammals

~p: There exists a goat which is not a mammal;
~p: There exists goats which are not mammals; ~p:

At least one goat is not a mammal.

Conditional Statement: Conditional proposition Given that p and q are propositions, the
compound proposition, "if p then q" which is symbolized p q is called a conditional proposition
or an implication. The connective is called conditional connective.
In the conditional proposition p q, p is called the antecedent or hypothesis while q is called the
consequent or conclusion.
Example 1
Consider the following propositions:
(i) If this month is January then next month is February. Here:

P: This month is January is the antecedent or hypothesis. q:

Next month is February is the consequent or conclusion.

(ii) If rain falls then I will wear a raincoat. Here:

P: Rain falls is the antecedent or hypothesis.

q: I will wear a raincoat is the consequent or conclusion.
The proposition if p then q has the following alternative meanings.
(i) p implies q.
Bi-conditional Statements:
Given that p and q are propositions, the compound proposition p if and only if q symbolized
is called the biconditional proposition. The bi-conditional proposition can also be
interpreted as: p is necessary and

sufficient for q. p implies q and q

implies p.

Contrapositive Statements:
If p is the statement 'a triangle is equilateral' and q is the statement 'a triangle is equiangular' the
statement ~ q ~p is 'if a triangle is not equiangular then it is not equilateral'. The statement ~ q
~p is called the contrapositive statement of p q
Further Readings: Argument.
Week 5: Logic 2

Objectives: the students should be able to

- Use the truth table to show tautology and contradiction
Proposition
Truth Table: Truth Table is used to perform logical operations in Maths. These operations
comprise boolean algebra or boolean functions. It is basically used to check whether the
propositional expression is true or false, as per the input values.
Construction of Truth Tables For:
OR AND, Conditional statement Biconditional Statement and contrapositive.
Further Readings are: Tautologies, and contradiction.

Week 6: Mapping and Functions

Objectives: the students should be able to

- Define mapping and function
- Distinguish the types of functions
- Solve problem involving mapping

A function is a relation between a set of inputs and a set of permissible outputs with the property
that each input is related to exactly one output. Let A & B be any two non-empty sets; mapping
from A to B will be a function only when every element in set A has one end and only one image
in set B.
Types of Functions
There are various types of functions in Mathematics, but we shall be limiting our studies to the
following function:

• Identical Function
• One-one Function (Injective Function)
• Onto Function (Surjective Function)

Part 7: Mapping and Functions

You will recall that a set is a clearly defined collection of objects, things or numbers. Given two
non-empty sets A and B, if there is a rule, which assigns an element x E A a unique element
y E B then such a rule is called a mapping. The set A is called the Domain of the mapping, while
the set B is called the Co-domain of the mapping.
Range: Is a subset of the co-domain, which is a collection of all the images of the elements of
the domain.

Identity mapping: The identity function is a function which returns the same value, which was
used as its argument.
x -2 -1 0 1 2
f(x) -2 -1 0 1 2

One to one mapping: An injective function (injection) or one-to-one function is a function

that maps distinct elements of its domain to distinct elements of its codomain.

Horizontal Line Test

A function is one-to-one if each horizontal line does not intersect the graph at more than one
point.

In the above example it only intersect the horizontal line only at one point. So f(x) is one-to-one
function which means that it has an inverse function.
Example 1:

Let A = {1, 2, 3} and B = {a, b, c, d}. Which of the following is a one-to-one function?
1. {(1, c), (2, c)(2, c)}

2. {(1, a),(2, b),(3, c)}

3. {(1, b)(1, c)}

The Answer is 2.
Explanation: Here, option number 2 satisfies the one-to-one condition, as elements of set
B(range) are uniquely mapped with elements of set A(domain).
Onto mapping:

Let f: X → Y be a mapping from the set X to the set Y. The mapping f is called an Onto-
mapping if every element of the co-domain is an image of at least one element in the domain
How to know if a function is onto?
You can compare the range and co-domain using the graph and determine whether the function
is onto or not. If the range equals codomain, then, function is onto.

Onto mapping is also referred to as Surjection

The following illustrations provide the clear difference between one to one, and onto function:

Inverse functions: An inverse function or an anti function is defined as a function, which can
reverse into another function. In simple words, if any function, “f” takes x to y then, the inverse
of “f” will take y to x. If the function is denoted by „f‟ then the inverse function is denoted by f-1.
One should not confuse (-1) with exponent or reciprocal here.
The inverse function returns the original value for which a function gave the output.
Example:
A function f on the set of real number is defined by determine
Solution:

Set , then make x the subject of the formula

Composite functions

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by g f,

is defined as the function g f : A → C given by g f (x) = g(f (x)), x A.
The figure below shows the representation of composite functions.

The order of function is an important thing while dealing with the composition of functions since
(f g) (x) is not equal to (g f) (x).
Example1: If and , then f of g of x, .
If we reverse the function operation, such as g of f of x,
Example 2

The function f is defined on the set of real numbers by Find and hence
determine: .
Solution:

. Hence,